27. Describe how you could calculate the sum of the 10^th to 20^th terms of the geometric sequence 3,6,12 24. __ using two different methods. Which way would you choose and why? (Do not find and sum each term separately as one of your methods)

Question

27. Describe how you could calculate the sum of the 10^th to 20^th terms of the geometric sequence 3,6,12 24. __ using two different methods. Which way would you choose and why? (Do not find and sum each term separately as one of your methods)

Answer 1

### 3144704

Answer 2

## Step 1: Finding the common ratio and the 10th term### The common ratio,

, is found by dividing any term by its preceding term.

. The

term of a geometric sequence is given by

, where

is the first term. Thus, the

term is

.## Step 2: Method 1: Sum of finite geometric series### The sum of a finite geometric series from term

to term

is given by

. In our case,

,

, and

. Therefore, the sum is

.## Step 3: Method 2: Difference of two sums### We can find the sum of the first 20 terms and subtract the sum of the first 9 terms. The sum of the first

terms of a geometric series is given by

. So,

. And

. The sum from the

to

term is

. There seems to be a calculation error in this method. Let's recalculate.

and

.

.## Step 4: Choosing the preferred method and correcting the error### Method 1, the direct formula for the sum of a finite geometric series, is more concise and less prone to errors. The discrepancy in Method 2 arises from a small calculation error. The correct calculation is:

. However, the sum of a finite geometric series from term m to term n is given by

. Thus,

.

Question

27. Describe how you could calculate the sum of the 10^th to 20^th terms of the geometric sequence 3,6,12 24. __ using two different methods. Which way would you choose and why? (Do not find and sum each term separately as one of your methods)

Solution

Answer

Explanation