Finite Geometric Sum Formula

This is a short, animated visual proof demonstrating the finite geometric for any ratio x with x greater than 1. This series (and its infinite analog when x is less than 1) is important for many results in calculus, discrete mathematics, and combinatorics.

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[JohnSlack]

The most amazing part of this is trying to pause the video before you visually explain it, not have a clue about how to do it, then seeing how "simple" it turns out to be. It's an amazing feeling actually to learn like this.

[vladimirrodriguez]

Fabulous animated geometric demonstration of this series. Congratulations and waiting for more material

[sambhavgiri]

Isn't it amazing how simple this is? Keep up the good work man!

[azorbz]

For a written explanation, let S = a+ax+ax^2+...+ax^(k-+1)

Note that (S-a)/x = a+ax+...+ax^(k-2), provided that x is non-zero (if x is zero, it is trivial to see that S = a)

Note also that S - ax^(k-1) = a+ax+...+ax^(k-2), which is the same as (S-a)/x

Equating the two gives (S-a)/x = S-ax^(k-1)

=> S-a = Sx-ax^k
=> S-Sx = a-ax^k
=> S(1-x) = a(1-x^k)
=> S = a(1-x^k)/(1-x)

Which is the same result as in the video.

This holds for all x ≠ 1. If x = 1, then it is clear that S = a+a+...+a =ka

[asparkdeity]

Maths is so beautiful, and u show it in the most elegant way possible 🥺

[davidbass]

Factor out a, consider sum as i goes from 0 to k-1 of x^i, call the value of this sum S, multiply both sides by X so that S*X = the sum as I goes from 0 to k-1 x^i+1
You get 2 equations, s = the sum as I goes from 0 to k-1 of x^i and the sum as I goes from 0 to k-1 of x^i+1
Subtract the two equations and move the terms of the second equation by 1 term
S-Sx = 1 +(x-x) + (x^2-x^2) + ... + (x^k-1-x^k-1) +(0-x^k)
S(1-x) = 1- x^k
S = (1-x^k)/(1-x)
Remember that we need to multiply by a since we factored it in the first step
Final answer:
a + ax + ax^2 + ... + ax^k-1 = (a-ax^k)/(1-x)

[aythanraherisoanjato]

Could we use a similar approach when x < 1 ? By removing area from a rectangle of width 1 ?

[mayankpawar]

Why we choose a/x-1 as the other side of the rectangle?

[su]

took me a while to understand that line. just because I did not see point "0"/origin. But nice

[thanhduyenphanthi]

well, i found and i used it for 1 year and first time i saw this but mutiplied by a

[sergche]

Indeed, using the same clever truck you can multiply the sequence by x-1 and see x terms cancelling out 🤔