Ideas in Geometry/Proofs by Picture
Section 2.4.3
Proofs Involving Sums
Formula for a sum of integers
[edit | edit source]1+2+3+4+5+...+n= [ n (n+1) ] /2
[edit | edit source]==== This picture is meant to show the pattern of infinite sums. If the height of the rectangle is "n", the width of the rectangle is "n+1". This then shows the bottom triangle's area, made up of the light colored circles, is equal to [ n (n+1) ] /2 Therefore proving for any number "n" it's infinite sum will be [ n (n+1) ] /2 ====
Adding up an infinite number of terms
[edit | edit source]Way to show a finite answer for a sum that continues "+...=___"
[edit | edit source]Pictures are used to show the sum.
[edit | edit source]==== Square shows that with each piece of the equation can fit into the next tiny section of the 1x1 square making the sum of all the pieces is 1 (1/2)+(1/2)^2+(1/2)^3+(1/2)^4+(1/2)^5+...=1
Triangle shows that the shaded region is each piece of the equation. It exaggerates with each layer created in the triangle, the shaded pieces take up (1/3) every time. This shows that the sum of all the pieces would be (1/3) (1/4)+(1/4)^2+(1/4)^3+(1/4)^4+(1/4)^5+...=(1/3) ====