SOLUTION: The sum of an infinite G.P. is 15 and the sum of the squares of the terms is 45. Find the G.P

Question 1122514: The sum of an infinite G.P. is 15 and the sum of the squares of the terms is 45. Find the G.P
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let the GP be

a, ar, ar^2, ar^3, ...

Then the GP with the terms squared is

a^2, a^2r^2, a^2r^4, a^2r^6, ...

The infinite sum of the GP is

(1)

The infinite sum of the squared GP is

(2)

Dividing (2) by (1) gives us

(3)

Then (1) and (3) give us two linear equations in a and r that we can solve:

(4)

Then substituting (4) in (1) gives us a=5.

The GP has first term 5 and common ratio 2/3:

5, 10/3, 20/9, ...

The squared GP is

25, 100/9 400/81, ...

CHECK:

The sum of the GP is

The sum of the squared GP is

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