Question

If the sum of an infinite GP a, ar, ar², ar³, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar², ar⁴, ar⁶, .... is ______.

Options

• `1/2`

• `2/5`

• `25/2`

• `9/2`

Answer 1

If the sum of an infinite GP a, ar, ar² , ar³, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar², ar⁴, ar⁶, .... is `underlinebb(1/2)`.

Explanation:

Given `a/(1 - r)` = 15  ...(i)

and `a^2/(1 - r^2)` = 150 ⇒ `(a/(1 + r))(a/(1 - r))` = 150

Using equation (i),

⇒ `a/(1 + r)` = 10  ...(ii)

Solving equation (i) and (ii), we get

`15/(1 + r) = 10/(1 - r)`

⇒ 15 – 5r = 10 + 10r

⇒ 5 = 25r

⇒ r = `1/5`

From equation (i),

a = 15(1 – r)

= `15(1 - 1/5)`

= `15 xx 4/5`

= 12

Now, ar² + ar⁴ + ar⁶ + .... + ∞

⇒ S_∞ = `(ar^2)/(1 - r^2) = (12 xx 1/25)/(1 - 1/5) = 1/2`