Sequences and Series Question

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The following algebraic work shows the derivation of the formula for the sum of a finite geometric series,

$S_n = a((1-r^n)/(1-r))$ . Answer the following questions to give the reasons or justification for each step in the work.

$(1) \ \ S_n = a + ar + ar^2 + ... + ar^(n-2) + ar^(n-1)$

$(2) \ \ S_n r = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n$

$(3) \ \ S_n - S_n r = a - ar^n$

$(4) \ \ S_n (1-r) = a(1-r^n)$

$(5) \ \ S_n = a ((1-r^n)/(1-r))$

Grade 11 Sequences and Series CCSS: HSA-SSE.B.4

The equation in the third step is found by subtracting the equation from step 2 from the equation in step 1. What happens to all the terms on the right hand side?

They are ignored, because they are all going to be much smaller than either $a$ or $ar^n$ , depending on whether $r$ is greater or less than 1.
All but two of them are eliminated by subtraction. Aside from the first term of the first equation and the last term of the second equation, each term in the first equation has an equal term in the second equation, and thus they become n - 2 zeros.
Using the factor theorem, they all cancel out except for $a$ and $ar^n$ .
Dividing both sides of the equation by $ar, ar^2, ... , ar^(n-1)$ , they all cancel out.

Question Info

Sequences and Series Question

Grade 11 Sequences and Series CCSS: HSA-SSE.B.4