Jack wants to buy a new computer for 35 000

Question

Jack wants to buy a computer for 35 000. He decides to save by depositing an amount of 500 once a month into an account earning 11.32% interest per year, compounded monthly. The approximate time it will take Jack to 35 000 available equals 1) 40 months 2) 54 months 3) 70 months 4) 115 months 5) None of the above

Answer 1

11.32% per year is 11.32/12=0.9433% per month. Let n be the number of months Jack needs to save over. The first month's deposit makes the most interest and accumulates according to the formula 500(1.009433)^n. The second month's deposit accumulates less: 500(1.009433)^(n-1). The final deposit accumulates 500*1.009433. The total accumulated over n months is the sum of these amounts, which form a series:

500*1.009433(1+1.009433+1.009433^2+...+1.009433^(n-1)

Let S be the sum of this series in brackets, then 1.009433S-S=0.009433=(1.009433^n)-1, so S=((1.009433^n)-1)/0.009433. So 500*1.009433*((1.009433^n)-1)/0.009433=35000.

504.7167((1.009433^n)-1)=35000*0.009433=330.167

1.009433^n=1+(330.167/504.7167)=1.6542

Taking logs: nlog1.009433=log1.6542, so n=log1.6542/log1.009433=54 months approx. (answer 2).