Sum of the terms of a geometric sequence

Verify if the sum 4^n - 1 represents the sum of the terms of a geometric sequence.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Let 4^n - 1 represent the sum of m terms of a GP. 4^(n - 1) - 1 is the sum of m - 1 terms of the GP.

Tn = 4^n - 1 - 4^(n - 1) + 1

=> 4^n - 4^n/4

For two consecutive terms Tn and Tn+1

Tn+1 / Tn = [4^(n + 1) - 4^(n + 1)/4]/(4^n - 4^n/4)

=> [4*4^n - 4*4^n/4]/(4^n - 4^n/4)

=> 4*(4^n - 4^n/4)/(4^n - 4^n/4)

=> 4

As we arrive at a common ratio between consecutive terms, 4^n - 1 represents the sum of terms of a GP.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial