Sum Of Infinite Terms A Ar Ar2 Ar3 So On Is 15 If The Sum Of The Square Of These Terms Is 150 Then Find The Sum Of Ar2 Ar4 Ar6 So On

Q. Sum of infinite terms A, AR, AR², AR³,…. is 15. If the sum of the square of these terms is 150, then find the sum of AR², AR⁴, AR⁶, ⋯.

Answer:

½

Solution:

A + AR + AR² + ⋯ = 15

⇒A/(1-R) = 15 ⋯(i)

Now, A² + A² R² + A² R⁴ + ⋯ = 150

⇒A²/(1-R² ) = 150

⇒(A⋅A)/((1-R)(1 + R)) = 150

⇒15A/(1 + R) = 150

⇒A = 10(1 + R)⋯(ii)

Solving (i) & (ii):

10(1 + R)/(1-R) = 15

⇒R = 1/5

So, A = 12

AR² + AR⁴ + AR⁶ + ⋯ = (AR^²)/(1-R² ) = (12⋅1/25)/(1-1/25) = 1/2