Question

PLEASE HELP!! 50 POINTS!! If you're not going to help, please don't answer.

Answer 1

Part A

A geometric sequence is where the terms increase by the same ratio.

Example:

7, 14, 28, 56, ...

We start at 7 and double each term to get the next term. The common ratio is 2.

============================================================

Part B

The next step is to subtract the two equations straight down. This will cancel the vast majority of the terms, and allow to solve for $S_n$ to get a fairly tidy formula. Refer to part C for more info.

============================================================

Part C

$S_n = a_1 + a_1r + a_1r^2 + \ldots + a_1r^{n-1}\\\\rS_n = a_1r + a_1r^2 + \ldots + a_1r^{n-1} + a_1r^n\\\\rS_n - S_n = \left(a_1r + a_1r^2 + \ldots + a_1r^{n-1}+a_1r^n\right)-\left(a_1 + a_1r + a_1r^2 + \ldots + a_1r^{n-1}\right)\\\\S_n(r - 1) = a_1r^n - a_1\\\\S_n = \frac{a_1r^n - a_1}{r-1}\\\\S_n = \frac{-a_1(r^n - 1)}{-(-r+1)}\\\\S_n = \frac{a_1(1-r^n)}{1-r}$

For more information about the canceling going on from step 3 to step 4, see the attachment below.