Question

1. If a person invests in a tax-sheltered annuity, the money invested, as well as the interest earned, is not subject to taxation until withdrawn from the account. In the questions below, assume that a person invests 20,000 each year in a tax-
sheltered annuity at 10 percent interest compounded annually. Let An represent the amount at the end of n years.
a. Find a recurrence relation for the sequence A0, A1,....
b. Find an initial condition for the sequence A0,A1,....
c. Find A1, A2 and A3.
d. Find an explicit formula for An.

2. 35, 000 is invested at 6.3% per annum compounding yearly.
a. How much interest was earned in the 3rd year of investment?
b. What is the investment worth at the end of 4 years?

Answer 1

Answer:

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1. If a person invests in a tax-sheltered annuity, the money invested, as well as interest earned, is not subject to taxation until withdrawn from the account. Assume a person invest $2000.00 each year in a tax-sheltered annuity at 10 percent interest compounded annually. Let € An represent the amount at the end of n years.

a. Find the recurrence relation for the sequence € A0,A1, A3,...

b. Find an initial condition for the sequence € A0,A1, A3,...

c. Find € A1, A2, and A3 .

Step-by-step explanation:

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Answer 2

Interest in 3rd Year = 2491.58

End of Year 4 -- 44 689.05

Explanation:

1.

We have the formula for Compound interest:

$C.I. \implies Amount = P (1+\frac{r}{n})^{nt}$

Therefore:

$A_0$ = 20 000

$A_1$ = 42 000

$A_2$ = 66 200

$A_3$ = 92 820

2.

We have  $C.I. \implies Amount = P (1+\frac{r}{n})^{nt}$

Therefore Amounts by Year:

End of Year 1 -- 37 205.00

End of Year 2 -- 39 548.92

End of Year 3 -- 42 040.50

End of Year 4 -- 44 689.05

Therefore:

Interest in 3rd Year = 42040.50 - 39548.92 = 2491.58

Interest in 4th Year = 44689.05 - 42040.50 = 2648.55