Find Final Amount on Deposit After 21 Years of Compounding Interest

[rain]

Sam deposts $10,000 at the beginning of each year for 12 years in an account paying 5% compounded annually. He then puts the total amount on deposit in another account paying 6% compounded semiannually for another 9 years. Find the final amount on deposit after the entire 21 year period.

Did I understand this question right?
He deposit 10,000 every year for 12 years and then put that amount in another account for another 9 years but without depositing anymore?

the answer i got is $270978.43

can anybody just check that for me? thanks.

 

[HallsofIvy]

I haven't memorized any "financial" formulas so let me see if I can think it through. Let A= 10000 (so I don't have to keep typing that!). Then the first A deposited sits in the bank for 12 years at 5% interest and is worth A(1.05)¹² at the end of those 12 years. The A deposited at the start of the next year is in the bank for 11 years and so have value A(1.05)¹¹. The A deposited at the start of the next year will have value A(1.05)¹⁰, etc. We add them all together and so have,at the end of the 12 years A(1.05)+ A(1.05)²+ A(1.05)³+ ...+ A(1.05)¹². That is a "geometric series" of the form \Sigma{n=0}^11 ar^n with a= A(1.05)= 10000(1.05)= 10500 (NOT A= 10000 because he did not add 10000 at the end of the last year) and r= 1.05. The formula for the sum of such a series is a\frac{1- r^{12}}{1-r}= 10500\frac{1- 1.05^{12}}{1- 1.05}= 10500\frac{-.7958}{-.05}= 167129.83
He will have $167129.83 after the 12 years (he deposited $120,000 himself and earned $47129.83 in interest).
Now, he deposits that into a bank at 6% interest compounded semi-annually for 9 years. Each half year, the money will have earned 3% interest and there are 18 half year periods in 9 years. At the end of the 21 years, the money will be worth 167129.83(1.03)¹⁸= $284527.35. That's slightly more than what you got!

 

[rain]

i got it now. thanks a lot. its very helpful.