Confused with some geometric problems

Mar 2010
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For the problem above part b) and c), what do they mean by "Find the expressions" and "Write the expression for Q_n in closed-form."?



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A bank account with a $75,000 initial deposit is used to make annual payments of $1000, starting one year after the initial $75,000 deposit. Interest is earned at 2% a year, compounded annually, and paid into the account right before the payment is made.

a) What is the balance in the account right after the 24th payment?


How would we find this out without actually calculating the problem 24 times. There must be a formula to this geometric problem.

I have tried the formula S_n = a(1-r^n)/1-r, with a = 75000m, r = 1.02, n = 26 but to no avail. The answer that it comes out to be is way to high to be reasonable. (Answer from the formula is 2.5 million)

What would be the correct formula to use in this problem to solve for the balance in the account right after the 24th payment? Thanks in advance.
 
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Grandad

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Hello florx
For the problem above part b) and c), what do they mean by "Find the expressions" and "Write the expression for Q_n in closed-form."?
They mean write an explicit formula; that is, one that doesn't involve things like infinite series, recursion, etc. Here's a definition; and here's an example using the Fibonacci sequence.

Having said that, I'm not sure what the answer is. I think you must assume that, initially, all of the \(\displaystyle 50\) mg dose is absorbed into the blood-stream. So \(\displaystyle Q_0 = 50\).

A bank account with a $75,000 initial deposit is used to make annual payments of $1000, starting one year after the initial $75,000 deposit. Interest is earned at 2% a year, compounded annually, and paid into the account right before the payment is made.

a) What is the balance in the account right after the 24th payment?


How would we find this out without actually calculating the problem 24 times. There must be a formula to this geometric problem.

I have tried the formula S_n = a(1-r^n)/1-r, with a = 75000m, r = 1.02, n = 26 but to no avail. The answer that it comes out to be is way to high to be reasonable. (Answer from the formula is 2.5 million)

What would be the correct formula to use in this problem to solve for the balance in the account right after the 24th payment? Thanks in advance.
If the number of years that has elapsed is \(\displaystyle n\), and the amount in the bank after the payment is made is $\(\displaystyle A\):

When \(\displaystyle n =0,\; A = 75000\)


When \(\displaystyle n =1,\; A = 75000\times1.02-1000\)


When \(\displaystyle n =2,\; A = (75000\times1.02-1000)\times 1.02-1000\)
\(\displaystyle =75000\times1.02^2-1000(1+1.02)\)
When \(\displaystyle n =3,\; A = (75000\times1.02^2-1000(1+1.02))\times 1.02-1000\)
\(\displaystyle =75000\times1.02^3-1000(1+1.02+1.02^2)\)
...

When \(\displaystyle n=24,\;A = 75000\times 1.02^{24}-1000(1+1.02+1.02^2+...+1.02^{23})\)


Can you complete it now, by summing the geometric series? (I make the answer $90,210.93.)

Grandad
 
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