Question

The sum to infinity of a geometric series is 4 and the second term is 1. Find the first,third and fourth terms.

Answer 1

2, 0.5 and 0.25.

Explanation:

The nth term an = a1r^(n-1) where a1 = first term and r = the common ratio.

The sum to infinity = a1/(1-r) for -1 < r < 1.

So we have:

Second term a2 = 1 = a1(r)(2-1) = a1r and

4 = a1/(1 - r)

From the first equation:

a1 =1/r

So substituting in the second equation:

4 = 1/ r / (1 - r)

4 = 1 / r - r^2

4r - 4r^2 = 1

4r^2 - 4r + 1 = 0

(2r - 1)(2r - 1) = 0

r = 0.5

So the first term a1 = 1/ r =1/0.5 = 2.

The third = 1*0.5 = 0.5

and the 4th = 0.5*0.5 = 0.25.