How to Solve Sum of Geometric Sequences

Question

[ I am learning about Geometric Sequences in my Algebra 2 class right now. While I found Arithmetic Sequences to be easy, this is more difficult and confusing. If I could get an explanation or something that would help me solve these problems, that would be terrific as I am going into Precalculus soon and need to be prepared. Thank you! ]

Example 1:

(-3) + (-3)² + (-3)³ + . . . + (-3)^{8 = ?}

Example 2:

9↑ ∑ ↓ i = 1

2 (1/2) ⁱ = ?

Answer 1

The main concept you need is the PARTIAL SUM.

This PARTIAL SUM is , in and of itself, another sequence, defined

in terms of common ratio r.

The first partial sum, denoted S1, is simply the first term. So S1 = 1 = r^0

The 2nd partial sum is S2 = 1 + r^1 = 1+r

THe 3rd partial sum is S3 = 1 + r + r^2

etc etc

The Kth partial sum is Sk = 1 + r + r^2 + r^3 + ... + r^k where k in integer greater than 3

The FORMULA to calculate ANY Kth Partial Sum is:

[ 1 - r^(k+1)]/ (1-r)

For example, to calculate the 10th partial sum, you do [ 1 - r^11]/(1-r)

So for your sequence , you have 3^k

So r = 3.

So the 8th partial sum is

[ 1 - 3^9]/(1-3) =

[ 1 - 19683]/(-2) =

-19682/(-2) =

9841