Prove the following formula for the computation of the future value of a stream of regular payments to be:
FV=R[((1+i)^n-1)/i];where R is the regular payment.
To prove the formula for the computation of the future value of a stream of regular payments, we can use the concept of compound interest.
Let's break down the formula and explain each component:
FV = R[((1+i)^n-1)/i]
FV: Future Value of the stream of regular payments.
R: Regular payment amount.
i: Interest rate per period.
n: Number of payment periods.
To understand the formula, we need to calculate the future value of each individual payment and then sum them up.
Step 1: Calculate the future value of each payment.
For the first payment:
Future value = R * (1+i)^(n-1)
For the second payment:
Future value = R * (1+i)^(n-2)
.
.
.
For the nth payment:
Future value = R * (1+i)^(n-n) = R
Step 2: Sum up the future values of all the payments.
Future Value (FV) = R * (1+i)^(n-1) + R * (1+i)^(n-2) + ... + R * (1+i)^(n-n)
This is an arithmetic series, and we can use the formula for the sum of an arithmetic series to simplify it:
Sum = R * [(1+i)^(n-1) + (1+i)^(n-2) + ... + 1]
Step 3: Apply the formula for the sum of a geometric series.
The sum of a geometric series is given by:
Sum = a * (r^n - 1) / (r - 1)
In our case, a = 1 and r = (1+i).
Therefore, the sum becomes:
Sum = 1 * ((1+i)^n - 1) / (1+i - 1)
= ((1+i)^n - 1) / i
Step 4: Multiply the sum by the regular payment amount R to get the future value.
FV = R * ((1+i)^n - 1) / i
Thus, we have derived the formula for the computation of the future value of a stream of regular payments:
FV = R * ((1+i)^n - 1) / i
This formula allows us to calculate the future value of a stream of regular payments based on the regular payment amount, interest rate per period, and the number of payment periods.