Question

How do you find `S_n` for the geometric series `a_1=4`, r=0.5, n=8?

Answer 1

`S_n = 7.96875`

Explanation:

there are two equations to find `S_n` of a geometric series.
if r is greater than 1, we use `S_n=a*(r^n -1)/(r-1)`
and if r is less than 1, we use `S_n=a*(1-r^n)/(1-n)`

in this question r is less than 1. so we use the second equation to find `S_n`

`S_n = 4* (1-0.5^8)/(1-0.5)`

therefore `S_n = 7.96875`

Answer 2

`S_8 = color(green)(7.96875)`

Explanation:

Given an initial value `color(red)(a_1)`, and a geometric increment of `color(blue)n`
the `color(brown)n^(th)` term is given by the formula:
`color(white)("XXX")S_color(brown)n=color(red)(a_1)((1-color(blue)r^color(brown)n)/(1-color(blue)r))`

Using the given values (and a calculator)
`color(white)("XXX")S_8 = 7.96875`