Question

For the following exercises, use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum.

Answer 1

Consider the following geometric sequence:

9Σn=1(5∙2n-1)

 

The explicit formula of above geometric sequence will be,

an = 5∙2n-1 ...... (1)

 

Substitute n = 1 in formula (1) and compute first term of above sequence,

a1 = 5 × 21-1

     = 5 × 1

     = 5

 

Substitute n = 2 in formula (1) and compute second term of above sequence,

a2 = 5 × 22-1

     = 5 × 2

     = 10

 

So, common ratio of above geometric sequence will be,

r = a2/a1

     = 10/5

     = 2

 

From the above summation, there are 9 terms in the above series. So, n = 9

Use the formula for the sum of first n terms of a geometric sequence,

Sn = a1(1 – rn)/(1 – r) ...... (2)

 

Substitute a1 = 5, r = 2 and n = 9 in formula (2) and simplify,

S9 = 5(1 – 29)/(1 – 2)

     = 5(1 – 512)/(1 – 2)

     = 5 × 511

     = 2555

 

Therefore, the sum of above series is S9 = 2555.

Therefore, the sum of above series is S9 = 2555.