Angle Between 2 Vectors

The a gle betqeen 2 vectors is a way to find the distance between the 2. It is simple plug and chug into the formula. 

Put all variables into the equation and solve. 

Binomial theroem

The binomial theroem is that an equation with a power can be expanded. This would be like (x-2)^4 and you would have to expand it out into the four parts to make one whole equation. The easiest way to do this is to use pasacals triangle. 

That is all that you have to do. Start on the left with the first term to the max power. On the last equation it would be to the 4. Then decrease from left to right. Then take the right term and do the same but start on the right and go left. You decrease the power every time. That is all

Rotation of conics

4 steps

1. Plug in to cot2ø= a-c/b

A is the number infront of the x^2 term b is infornt of the xy term and c is infront of the y^2 point. 

2. Solve for x and y prime by plugin in for x=xcosø-ysinø and y=xsinø-ycosø

3. After that substitute in the original equation. 

4. Simplify and done

Determinants of Matrices

A determinant of a matrix is a constant number for a matrix. A matrix is a set of numbers in a set of brackets. 

Shortcut of 3x3 is to take the first to columns and copy them on the outside of the matrixs then take the diagonals and multiply. 

Dot product

The dot product is a constant number for any two or more vectors. A vector is a line wiht direction and magnatuide. What you do is times the x1andx2 for you first x. Then y1 and y2 for y variable. Lastly would be the z1 and z2 to get z. A dot product is the line between the two vectors. Then add all of those together

Parametric equations

A parametric equation is where the graph goes over time. The whole point is to eleminate the parametric variable t. This jusat envolves solving for x then pluging the t variable into y. 

X=t+1 y=t-1

X-1=t 

Y=x-2

Parametric graph. 

That is all

Tangnt lines

Tangent lines are a line that comes in contact with a point and has its own slope. 

Xt is slope of a graph. As explained earlier in the pictures it is just rise over run or (f(x+h)-f(x))/h. This allows one to find the tangent line. One term that someone would come across is a deritive which is really just the slope of a line or the previous equation. 

Next is the short cut to this process because it is long and lengthy. It is simple. Take the power of the variable move it down and multiply the constant by that number then decrease the power by one. Do this for each variable. On a constant without a varible will cancle out. 

Random trivia

Triskaidekaphobia means fear of the number 13. Paraskevidekatriaphobia means fear of Friday the 13th (which occurs one to three times a year). 

If you sneeze too hard, you can fracture a rib. If you try to suppress a sneeze, you can rupture a blood vessel in your head or neck and die. If you keep your eyes open by force, they can pop out. (DON'T TRY IT)

Every three seconds a baby is born somewhere in the world. 

On average, 12 newborns will be given to the wrong parents every day. 

Most lipstick contains fish scales 

The most common name in the world is Mohammed. 

When you die your hair still grows for a couple of months. 

The most money ever paid for a cow in an auction was $1.3 million. 

The sound of E.T. walking was made by someone squishing her hands in jelly. 

There are 10 human body parts that are only 3 letters long (eye hip arm leg ear toe jaw rib lip gum). 

A skunk's smell can be detected by a human a mile away. 

The human heart creates enough pressure to squirt blood 30 feet (9 m). 

When snakes are born with two heads, they fight each other for food. 

The average person makes about 1,140 telephone calls each year. 

Stressed is Desserts spelled backwards. 

If you had enough water to fill one million goldfish bowls, you could fill an entire stadium. 

Charlie Brown's father was a barber. 

Dentists have recommended that a toothbrush be kept at least 6 feet (2 m) away from a toilet to avoid airborne particles resulting from the flush. 

You burn more calories sleeping than you do watching TV. 

A lion's roar can be heard from five miles away. 

The average person spends about 2 years on the phone in a lifetime.

The largest number of children born to one woman is recorded at 69. From 1725-1765, a Russian peasant woman gave birth to 16 sets of twins, 7 sets of triplets, and 4 sets of quadruplets. 

The roar that we hear when we place a seashell next to our ear is not the ocean, but rather the sound of blood surging through the veins in the ear. Any cup-shaped object placed over the ear produces the same effect 

The United States has never lost a war in which mules were used. 

Children grow faster in the springtime. 

On average, there are 178 sesame seeds on each McDonalds BigMac bun. 

7.5 million toothpicks can be created from a cord of wood. 

The plastic things on the end of shoelaces are called aglets. 

About 200,000,000 M&Ms are sold each day in the United States. 

There are no clocks in Las Vegas gambling casinos. 

The numbers "172" can be found on the back of the US 5 dollar bill, in the bushes at the base of the Linc 

Women blink nearly twice as much as men. 

Every year approximately 2,500 left-handed people are killed by using object or machinery designed for right-handed people. 

Turtles can breathe through their butts 

It’s estimated that at any one time around 0.7% of the world’s population is drunk. 

More people are kill by falling coconuts every year then sharks attack 

Mel Blanc (the voice of Bugs Bunny) was allergic to carrots 

The only 15 letter word that can be spelled without repeating a letter is uncopyrightable. 

Forty percent of Americans iron their clothes while wearing their underwear or being completely naked. 

If you eat enough carrots, your skin will eventually turn orange. 

Porcupines float in water. 

Bookkeeper is the only word in the english language that has 3 letters that consecutively repeat. 

Ants do not sleep. 

Between 1902 and 1907, the same tiger killed 434 people in India. 

315 entries in Webster’s 1996 dictionary were misspelled. 

The winter of 1932 was so cold that Niagara Falls froze completely solid. 

I am. is the shortest complete sentence in the English language. 

Annually Americans eat 45 million turkeys at Thanksgiving. 

Los Angeles’ full name is “El Pueblo de Nuestra Senora la Reina de los Angeles de Porciuncula.” 

The Ramses brand condom is named after the great pharoh Ramses II who fathered over 160 children. 

There are 66 million different possible pizza combinations on the Domino's pizza menu. 

Americans, on average, spend 18% of his or her income on transportation as compared to only 13% spent on food. 

A giraffe can lick it's own ears. 

Ketchup was sold in the 1830s as medicine. 

The most overdue book in the world was borrowed from Sidney Sussex College in Cambridge, England and was returned 288 years later. 

A giraffe's heart weighs an incredible 24 pounds. 

Walt Disney was afraid of mice. 

The word “lethologica” describes the state of not being able to remember the word you want. 

The original Coca Cola was green in color. 

It would take 1.2 million mosquitoes biting you simultaneously to drain all your blood. 

No piece of paper can be folded in half more than 7 times. 

Donkeys kill more people annually than plane crashes. 

Canadians drink more coffee than anyone else in the world. 

Evaluating Limits

A limit is the point at wich two lines will intersect and stop or a point wich a line will no longer increase or decrease. They can neither continue on because both lines are in eachothers way. Limits tell us where we can start a graph but we need to know how to find them. That is where 3 simple rules come into play. 

These all tell us how to find our limits. But wait there is a trick. You must start with substitution to begin with. Trust me it will make you life much easier. If you plug in a number and get a nember that is not 0/0 then that is your limit. If it is 0/0 then that is the intermediate form and you have to use the other two methods. 

CANCELLLATION TECHNIQUE

This is where you try to cancle out a part of an equation from either the top or bottom so it will no longer equal 0. This is just factoring and solving. Here are two examples

The next technique is more confusing and harder. 

RATIONALIZATION TECHNIQUE

This is when there is a radical in the equation. First check to make sure there is an intermediate form. If there is then you must find the conjugate of the rational. If it is 4+rad8 then the conjugate is 4-rad8. Here are a couple examples. 

Now there is a tech ique to find out what the limit is if two lines are approaching one another and either have a pont for a limit or do not have limits. This is where one uses the one sided limit form. 

Here is an example of this. 

Distance, midpoint, and sphere formulas

Today we will learn about the formulas for Distance, Midpoint, and Spheres. 

The distance is quite simple because you take your points and you take the square root of all the points (x2-x1)^2+(y2-y1)^2 and so on. You the. Take the square root of the entire equation. That easy kinda of the same with midpoint. Here are the two formulas.

Pretty simple now into spheres. The equation is simple if all points are known. The center is (h,k,j)

You then put it into the formula (x-h)^2+(y-k)^2+(z-j)^2=r^2. That is it. Just plug you points and find the answer. Simple

Cross products

A cross product is used to find the cross between two matrices. It is written as UxV. The cross product is used to find the area of parallelograms. 

Cross products are quite simple but you can make it even simpler. How? By taking the determinant of the matrix instead of all that other work. Much easier. 

Lastly is the area of parallelograms and triple scaler or vollume of parallelpiped

Squaring numbers between 60 & 40

Here's a quick way to square numbers between 40 and 60. You know the squares of 40, 50, and 60, right?

So let's show what to do for N=51 through 59. Write N as (50+R) for R between 1 and 9. Then N² will have first two digits (25+R) and last two digits R². Thus:

52²= 2704,

58²= 3364,

since 27=25+2 and 33=25+8.

The rule for N=41 through 49 is similar. Write N as (40+R) for R between 1 and 9. Then N² will have first two digits (15+R) and last two digits (10-R)². So:

42²= 1764.

48²= 2304.

3D Graps

Conic sections on a Polar Graph

many people wonder what the heck is a conic on a polar? This is putting the redular shapes ofarabola, elipse, and hyperbola on a polar plane. It is simple yet effect but there are a few points that must be figured out first.

These are simple equations that must be memorized in order to graph a conic. Now take these simple equations and lets figure out how to graph it. Alough you still need more information. What? Yah ypu need to know how a graph will look just based off of the equation. It will save time instead of plug and chugging multiple points that takes a ton of time. Now no one has that one their hands so here are a few simple rules. 

That is pretty much all there is. Here is a sample problem worked. Find the eccentricity of the equation and what shape it is?

That is it. Enjoy the fun of a graph. 

Math is fun.

Her is a link to a math game that can pass hours of time in minutes. Literally. 

http://www.playkidsgames.com/games/mathfact/mathFact.htm

Polar coordinates

A polar coordinate is complicating the regular graph of the x,y plane. Polar coordinates are now radiuses and angles. It helps to know some conversion factors between the x,y plane and the polar one. 

These conversion factors allow one to go from regularnt complex. Now what doesnthe graph look like?  It is just a bunch of circles and lines but one must know their unit circle to know how to use the graph. 

You count out the circles by your r. So if it is 2 in the r place count put 2 circles. After that you look at the theta to see the angle such as pi/4 and so one. 

That is all just plug and chug and graph. Yay fun. 

Fun facts of the day

• The numerical digits we use today such as 1, 2 and 3 are based on the Hindu-Arabic numeral system developed over 1000 years ago.

• Different names for the number 0 include zero, nought, naught, nil, zilch and zip.

• The smallest ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

• 2 and 5 are the only prime numbers that end with a 2 or a 5.

• The golden ratio of approximately 1.618 between two quantities such as lengths often appears in nature (tree branching, uncurling ferns, pine cone arrangements etc) and has been used throughout history to create aesthetically pleasing designs and art works such as Leonardo da Vinci’s Mona Lisa.

• Fibonacci numbers are named after Italian mathematician Leonardo of Pisa (better known as Fibonacci) who introduced them to Western Europe after they had earlier been described by Indian mathematicians. They are related to the golden ratio and proceed in the following order: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, .... Can you see the pattern?

• The number Pi (the ratio of the circumference to the diameter of a circle) can’t be expressed as a fraction, making it an irrational number. It never repeats and never ends when written as a decimal.

• Here is Pi written to 100 decimal places:
  3.1415926535897932384626433832795028841971693993751
  058209749445923078164062862089986280348253421170679

• What comes after a million, billion and trillion? A quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion.

• The name of the popular search engine ‘Google’ came from a misspelling of the word ‘googol’, which is a very large number (the number one followed by one hundred zeros to be exact).

• A ‘googolplex’ is the number 1 followed by a googol zeros, a number so ridiculously big that it can’t be written because there literally isn't enough room in the entire universe to fit it in!

• Check out some more big numbers.

• You might have heard the word ‘infinity’ before or seen its symbol that looks like the number 8 placed on its side. Infinity means a limitless quantity or something that goes on forever. While it’s not really a number like 1, 2 or 3, infinity is often used in math as part of equations and formulas.

• 111111111 x 111111111 = 12345678987654321

• 12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

• 
  

Elipses

What is an elipse? It is the sum of all points whose distance from two distinct fixed points, foci, is constant. 

this video explains everything to know about an elipse

► 6:51► 6:51www.youtube.com/watch?v=pLXR0yHmUmg

Parabolas

What is a parabola? Well it is the set of points that have the same distance from the focus and directerix of one other point.

Key things to remeber when solving for Parabolas. 

1 Solve for the square

2 when solving rember to take the second term (bx) and divide the b by 2 then square

That is all. 

Math Changes your View on the World

Trying to make math cool is like trying to make "not holding your breath for five minutes" cool: It already is, and anyone thinking otherwise struggles through life with significantly reduced mental abilities. Anyone born into a world based entirely on numbers with little dollar signs in front of them, then deciding they don't need to know numbers, is volunteering to be a very small one adding up to someone else's very big one.

"Just 10 more short years, and you'll have that degree in Klingon paid off!"

But accounting is to math what diapers are to biochemistry: dealing with the stinking mess left by some unfortunate and immature side effects of being alive while doing it. Mathematics isn't cool in the same way spacetime isn't real estate. It's much bigger and more important than the ridiculous little structures we've erected on it. And anyone who doesn't understand how truly cool it is (2.725 K) simply doesn't appreciate the sheer scale of it.

Ymir's autopsy

We found reality's programming language. Physics is the operating system, but it's written in equations. E=mc^2 defines large parts of reality in fewer bytes than it takes to say "my penis" and has similar effects on local spacetime, ladies (~ 90 percent, + gentlemen ~ 10 percent). But that's physics. There are pure mathematical equations which are just as capable of blowing your mind.

#5. Infinite Pi

Pi is the ratio of a circle's diameter to its circumference, in the same way nuclear fission is a way of powering TVs to watch America's Got Talent: an appallingly simple effect of a reality-defining truth. Pi isn't a number, it's a startup constant of spacetime. Take a line in one dimension, rotate it around another, and the resulting ratio of lengths is a precise number. The existence of space has a numerical signature. It's called a transcendental number, because even attempting to think about how much it means is more mind-expanding than all the drugs.

Seen here in the three dimensions it glues together

Calculating pi has become the computer scientist equivalent of tuning muscle cars: we don't actually need more digits for anything useful, because the basics do everything we actually need it for, but we've spent years stacking up air-cooled hardware just because. In 1985 it was calculated to 17 million digits. Srinivasa Ramanujan found the formula used. Around 1910.

It wasn't the only such formula, but was incredibly useful because it converged exponentially compared to other algorithms, making it ideal for computers. Interesting note: at the time there was no such thing as computers. Srinivasa Ramanujan had pre-empted processors by decades. In 1985 his formula was used in the world record calculation of pi to 17 million digits, and a slight variant modified by David and Gregory Chudnovsky now holds the title at 10 trillion. We're not saying he's a robot, but if a perfect calculating machine ever goes missing in a time travel experiment we already know where it ended up.

Konrad Jacobs

It turns out that rogue Terminators like sharp suits and counting forever.

#4. Dividing By Zero

Dividing by zero gives you undefined, infinity, or who cares, depending on whether you're talking to a mathematician, physicist, or engineer. You can tell a lot about how disconnected someone is from reality by how they contemplate infinity. For example, cult leaders often explain how infinity after nothingness means you should obey them.

"Pay your entire life now for insurance against eternity with the King! No refunds!"

At its most basic, division means, "How many times can you take this out of that?" For example, you can take a human head from the members of One Direction five times before you get blessed silence. On the one hand you can take nothing out of something an infinite number of times, but on the other, no, you can't, because that would take forever. It's a mathematical problem so hard it once crippled a U.S. Navy cruiser. The USS Yorktown's experimental military computer control system tried it and was crippled by a buffer overrun. Making this the first ship to overflow without any water. On the upside, it's way easier than asking a computer to define love or telling it that truth is a lie, or asking if those are both the same question.

"These feelings are tearing me apart! Also the incoming torpedoes."

The solution is as brilliant as it sounds stupid. You deal with the impossible by sneaking up on it. It's nice to know calculus solves problems the same way SEAL teams do it: sneakily and permanently.

Taking the limit as X goes to zero means that you never actually get there, but you get as close as you need to be, no matter how close that is. Then the "limit" as you tend toward zero -- but never actually reach it -- gives you the answer. As X gets small, sin X is approximately equal to X, so you're always dividing something by itself and getting one. Then, with a particularly cunning flourish, you end up dividing nothingness by itself and becoming one. Which I think means mathematics is the king of zen.

His decades of calm are going to be wasted when he finds out mathematicians worked it out with a pencil.

There's a fairly easy proof using the squeeze theorem. Which doubles as a good line when you want to chat up a mathematician.

"... and you'll be with me because I give quite excellent dick, QED."

These limits as you hit zero, which is a much less depressing sentence in mathematics, are essential for calculus. And calculus is essential for everything. If math is the programming language of reality, calculus is the graphics processor working out things like explosions, lasers, and gravity, all the cool special effects. Or as we call them in physics, effects.

#3. Pick A Digit, Any Digit (of Pi)

If the Ramanujan formula was transcendent understanding of reality by the human brain, the Bailey-Borwein-Plouffe formula is outright sorcery. Developed by Simon Plouffe in 1995, this formula lets you skip straight to any digit of pi without working out the rest of the number.

You want the the 10-trillion-and-33rd digit of pi? No problem. There's a fundamental constant of reality and now, without any input apart from being in the same reality, this equation can read out any part of it. That's like pulling rabbits out of a hat where the rabbits are Star Trek energy beings and your hat is a beret.

The process of using the formula sounds like someone found a glitch in god's computer. If you want the Nth term, you split the infinite sum at the Nth term, and a bit of modulo math skims out the required digit in hexadecimal. That sounds like something you'd read on Mr Mxyzptlk's GameFAQ. Reality, mathematics, and the design of 8-bit computers lining up to accidentally output one of the universe's BIOS settings. The craziest part? It's still slower than Ramanujan for finding the whole thing. It really is just a cheat code for reality.

Binomial Expansion

Today we will learn how to expand a binomial into its fully expanded form. There are 2 ways. 

One way is the binomial therom

(A+b)^n

All you do is foil the entire thing to the nth time. Yah its a ton of work no fun. 

The other way is to use pasacals triangle. Much more fun cause it is a ton easier. 

A lot easier. How? Well you see that row 0,1,2... And so on well that corresponds to the n power of your equation. 

How do you do this. You take the coeffiecents that are needed then you multiply by the correct term. How do you eo this? Well it is quite simple. First take your right hand term and take the highest power and multiply the first coefficent by it. Next you move to the term right next to it and subtract one from the n power. Thats you right hand side. Next is your left hand side. All you do is the exact same just start at the right side and move to the left. After you have expanded simplify and you have your answers simple right. 

Here is an example. 

Well ordering principle

Well-ordering principle: Every nonempty subset T of N has a least element. Thatis,thereisanm∈T suchthatm≤nforalln∈T.

Intutively clear as it may seem at the first glance, this principle turns out to be logically equivalent to the mathematical induction, the fifth axiom of Peano, which is quite surprising.

Theorem 1. The mathematical induction is logically equivalent to the well-ordering principle.

Proof. Part I. We show the well-ordering principle implies the math- ematical induction.

LetS⊂Nbesuchthat1∈Sandk∈Simpliesk′∈S. Wewant to establish that S = N by the well-ordering principle.

Suppose N\S is not empty. Then by the well-ordering principle there is a least element m ∈ N\S. Since 1 ∈ S we know 1 ∈/ N\S. Therefore m ̸= 1 and so by one of the homework 2 problems there is some q ∈ N such that m = q′ = q+1, which implies q < m by the definition of <. Weconcludethatq∈S;orelsewewouldhaveq∈N\Sandso m would not be the least element of N \ S, which is absurd. However, since S has the property that k ∈ S implies k′ ∈ S, we conclude that m = q′ ∈ S because q ∈ S. This contradicts m ∈ N \ S.

The contradiction establishes that N \ S is empty. Hence S = N. Part II. We show the mathematical induction implies the well-ordering principle.

Let S(n) be the statement: Any set of natural numbers containing a natural number ≤ n has a least element. Consider the set

E ={m∈N:S(m) is true}.

1 ∈ E because 1 is the least element of N (why?).

Wenextshowm∈Eimpliesm′∈E. Nowm∈EmeansifXis a subset of N containing a natural number ≤ m, then X has a least element. From this we want to establish m′ ∈ E.

So let C be any subset of N containing a natural number ≤ m′. If C has no element < m′, then m′ is the least element of C and we are done. Otherwise, we can now suppose there is a natural number y ∈ C such that y < m′. In particular, y ≤ m because by one of the homework 3 problems we know there is no natural number strictly between m and m′. Therefore, C now has an element y ≤ m, so that the induction hypothesis given in the preceding paragraph implies that C has a least element. In any event, we have proven C has a least element, so that

m′ ∈ E. Hence, the mathematical induction implies E = N.

 In summary, the mathematical induction implies that the statement S(n) is true for all n ∈ N.

To establish the well-ordering principle, let T ⊂ N be a nonempty set. There is some m ∈ T because T is nonempty, so that T contains a natural number ≤ m, which is just m itself. Then T has a least element because the statement S(m) is true by the preceding paragraph 

Amaze the Teach

 
Who doesn't love magic tricks? Here, we're going to look at how you can amaze your friends with some simple tricks involving numbers. You don't need a long-sleeve shirt, a rabbit, or even know the slightest bit of magic – just a little math.

Let's start off with a simple math trick.

Lightning Addition
Proclaim to your friends: “I'm a human calculator! I can add five 3-digit numbers quicker than you can punch in the numbers in a calculator.” Then ask three friends to write down three separate and random 3-digit numbers. For instance:

240
520
842

Ask them to return the paper, upon which you add in the final two 3-digit numbers.

240
526
842
759
473

Race against your friends, who are armed with a calculator, to add all these numbers up, while you blurt out the answer within a matter of seconds: 2,840.

The secret: The last two numbers you throw in isn't random at all – it may look random to your friends, but you've carefully chosen them so that the fourth and first numbers add up to 999 (in the example above, it would be 759, since 240 + 759 = 999). Choose the fifth number so that it adds with the second number to give 999 (473 + 526 = 999).

Once you have that down, the rest is simple: the sum is 2,000, add the number in the middle, and subtract 2. And voila! 2,840.

Lightning Addition 2
If your friends weren't too impressed by that trick, try this one. Ask a friend to pick two different single-digit numbers (for example, 4 and 7) and add them up (4 + 7 = 11). Tell them to do it in secret, so that you can't see what numbers they chose, or what they're doing.

Now, ask them to take the sum of the first two numbers (11), and add it to the number prior to that (which is 7). So the chain of calculations now appears to be as such:

  4
+7  
+11
18

Continue on the chain of addition until he reaches 10 numbers.

  4
+7    
+11  
+18  
+29  
+47  
+76  
+123
+199
+322
????

Once done, ask your friend to show you the chain of numbers. Here comes the amazing part: challenge your friend, who's armed with a calculator, to figure the sum of all these 10 numbers, while you figure out the answer in less than five seconds – which is 836.

The secret: It turns out that this pattern of addition follows the Fibonacci sequence of numbers, which is the basis for the Fibonacci ratio. You might know it as the Golden Ratio, which often occurs in nature.

The trick here is that the seventh number in the sequence, multiplied by 11, will always equal the sum of the 10-number sequence. In this case, it's 76 x 11 = 836.

Wait a minute, you say. How does one multiply the number by 11 in under five seconds? Here's where you use another arithmetic trick: Take the two numbers, 7 and 6, and split it, leaving a blank space in the middle:

7 __ 6

In the space between, place the sum of the two numbers:

7 13 6

Because the space in between 7 and 6 can only have one digit, carry forward 1 to 7, and what you end up with is:

836

Simple? Try it with 48 x 11

4 __ 8

Place the sum of the two numbers

4 12 8

Then, carry forward the 1 to the 4

528

Note: If your friend starts off with 9 and 7, or 9 and 8, the seventh number would be a 3-digit number (101 and 109, respectively). In these special cases, just remember that the result after multiplying them by 11 is 1,111, and 1,199 respectively.

Arithmetic Sequences

What is an arithmetic sequence? It is when an identifiable difference can be found between two different terms. 

2,4,6,8

in this sequence the difference between wech term is +2 or the variable d. D is used to find the formula of the sequence.

next is how to find the sum of an entire sequence without finding each and every variable. this is accomplished by n/2(A1+An)

Sums with sigman notation

What is a sum? It is the combination of all terms added together. The easiest way to complete this is by sigmun notation. Although it is the easiest way there is some calrification that is needed such as what are the upper and lower limits. 

Now that we know what the signs are, how do we do this? You start with the first term or A1. the 1 is the variable n so you would plug it in to the formula. The n at the top is the upper limit. DO NOT PASS IT. 

lastly there are a few rules that must be applied.