How much should she compute the required annual investment?

Question

A business woman wants to invest a certain amount of money at the end of each year for 5 years. The investment will earn 6% compounded annually. At the end of 5 years she will need a total of $30,000 accumulated. How much should she compute the required annual investment?

Answer 1

The easiest way to calculate these, is to calculate each payment and it's interest individually. There are 5 payments:

Step 1: Write the equation:
30,000 = x * 1.06^4 + x * 1.06^3 + x * 1.06^2 + x * 1.06 + x

Step 2: Rewrite with parenthesis:
30,000 = x * (1.06^4 + 1.06^3 + 1.06^2 + 1.06^1 + 1)
30,000 = x * (1.26247696 + 1.191016 + 1.1236 + 1.06 + 1)
30,000 = x * 5.63709296
30,000 / 5.63709296 = x
x = $5321.89 (solution!)

Another way to do it is: x * ((1+r)^(t+1)-1)/r, where r is the rate, and t is the number of terms.
30,000 = x * ((1.06)^5 - 1)/.06 = x * .338225578/.06 = x * 5.63709296