Interest Rate IRA

Question

Bob makes his first 1600 deposit into an IRA compounded annually on his 24th birthday and his last deposit on his 45th birthday. With no additional deposits the money in the IRA continues to earn 6.5 interest compounded annually until Bob retires on his 65th birthday. How much is in the IRA when he retires?

Answer 1

If the annual interest Is I = 6.5% = 0.065, the deposit d =1600 grows after n years to d_n = d(1 + I)ⁿ. At age 65 each annual deposit d will be hold for (65 - 45) = 20 or (65 - 44) = 21 or 22, 23.... or (65 - 24)=41 years. The accumulated sum D will be

D = d(1 + I)²⁰ + d(1 + I)²¹ + d(1 + I)²² +......+d(1 + I)⁴¹ = d(1 + I)²⁰ ∑_k=0 ^k=21 (1 + I)^k.

This is the geometric series and thus

D = d(1 + I)²⁰[1 - (1 + I)²²]/[1 - (1 + I)]= d(1 + I)²⁰ [(1 + I)²² - 1]/I

or D = 1600x1.065²⁰x(1.065²² - 1)/0.065 = 259,913.28.