Question

At the beginning of each month an amount of

 

X rand is deposited into a savings account earning k × 100%

interest per year, compounded monthly. The future value of these savings after 24 deposits can be denoted

by

Answer 1

The monthly interest rate is k/12. When X is deposited on the first month, it grows to X(1+k/12) after a month. On the next deposit the amount in savings becomes X(1+k/12)+X and at the end of the second month the amount grows to X(1+k/12)^2+X(1+k/12) and the next deposit added to make X(1+k/12)^2+X(1+k/12)+X. After n months we have X(1+k/12)^n+X(1+k/12)^(n-1)+...+X(1+k/12)+X=X((1+k/12)^n+...+1). This is a geometric progression we'll call S so we have here XS. (1+k/12)S-S=(1+k/12)^(n+1)-1, so S=12((1+k/12)^(n+1)-1)/k and after 24 deposits (2 years) X becomes XS=12X((1+k/12)^(n+1)-1)/k.