Question

What is the sum of the first five terms of a geometric series with a1 = 10 and r = 1/5?

Express your answer as an improper fraction in lowest terms without using spaces.

Answer 1

`(1562)/(125)`

Explanation:

We are given that

`a_1=a=10, r=(1)/(5)=0.2`

We have to find the sum of first five terms of geometric series.

We know that nth term in geometric series is given by

`a_n=ar^(n-1)`

`a_2=10((1)/(5))=2`

`a_3=10((1)/(5))^2=(2)/(5)`

`a_4=ar^3=10((1)/(5))^3=(2)/(25)`

`a_5=ar^4=10((1)/(5))^4=(2)/(125)`

`S_5=a_1+a_2+a_3+a_4+a_5`

Substitute the values then, we get

`S_5=10+2+(2)/(5)+(2)/(25)+(2)/(125)=(1250+250+50+10+2)/(125)=(1562)/(125)`

Hence, the sum of first five terms in geometric series =
`(1562)/(125)`

Answer 2

a 1 = 10
a 2 = 10 * 1/5 = 2
a 3 = 2 * 1/5 = 2/5
a 4 = 2/5 * 1/5 = 2/25
a 5 = 2/25 * 1/5 = 2/125
S 5 = a 1 + a 2 + a 3 +a 4 + a 5= 10 + 2 + 2/5 + 2/25 + 2/125 =
= 1250/125 + 250/125 + 50/125 + 10/125 + 2/125 = 1562 / 125