Annuities- The accumulated amount after 8 years

Question

Calculating annuities. The accumulated amount after 8 years of monthly payments of 1900 each into an account earning 9,7% interest per year, compounded monthly, equals?

Answer 1

8 years of monthly payments is 96 payments of 1900. Compound interest rate monthly is 9.7/12=0.8083% per month. The first monthly payment accrues to 1900(1.008083)^96; the second payment becomes 1900(1.008083)^95, and so on, up to the last payment which presumably gains only a month's interest.

So we have a series,S=

1900((1.008083)^96+1900(1.008083)^95+...+1900(1.008083)^2+1900(1.008083))=

1900(1.008093+...+1.008083^96).

S=1900*1.008083(1+...+1.008083^95). Let's call the bracketed series s. Multiply s by 1.008083: (1.008083+...+1.008083^96), so 1.008083-1.008083s=0.008083s=1.008083^96-1, so we can find s=

(2.166-1)/0.008083=144.247. And S=1900*1.008083s=276285.50.