Question

What is the sum of the first five terms of the geometric sequence in whicha1 = 3 and r= 1/3?Express your answer as an improper fraction using the slash (/) key and nospaces.

Answer 1

To solve this, we can use the formula for the sum of the first n terms of a geometric sequence. The fromula is:

`S_n=a_1((1-r^n)/(1-r))`

In this case, a1 = 3 and r = 1/3

Then:

`S_n=3((1-((1)/(3))^n)/(1-(1)/(3)))`

And to find the sum of the first 5 terms, we evaluate for n = 5:

`S_5=3((1-((1)/(3))^5)/(1-(1)/(3)))=3((1-(1)/(243))/((2)/(3)))=3(((242)/(243))/((2)/(3)))=3((3\cdot242)/(2\cdot243))=3\cdot(726)/(486)=3\cdot(121)/(81)=(121)/(27)`

Thus, the answer is:

`S_5=(121)/(27)`