## Series Application / Induction Exercise

1. \$7000 is invested at 10% per annum for 5 years with the interest compounded every 3 months (quarterly)

How much is the investment worth at the end of the 5 year period?

2. A woman pays \$500 into a retirement fund at the beginning of each year. The fund pays 8% per annum each year for 20 years. How much is in the fund at the end of the 20 years?

3. A couple take out a \$150 000 mortgage to be repaid in 20 years. The annual interest rate is 7.5% compounded monthly.

a. Find the monthly repayments

b. How much interest was paid in total

c. How much of the original mortgage is outstanding after 15 years?

4. Use induction to prove these statements:

a. 12 + 22 + ..    + n2 = 1/6.n(n + 1)(2n + 1)

b. 13 + 33 + ..      + (2n − 1)3 = n2(2n2 − 1)

c.

d. 32n − 1 is always divisible by 4

e. n3 + 6n2 + 2n is always divisible by 3

5. Show that if the statement:

S(n): 1 + 3 + 5+ .. (2n − 1) = n2 + 4

is true for n = k. It is also true for n = k + 1.

Does this mean it is true for all positive integers n? Explain.

6. Given

a. Obtain an expression for the coefficient of xn in the expansion of (1 + x)n(1 + x)n

b. Using the fact that nCk = nCn − k and noting that (1 + x)n(1 + x)n = (1 + x)2n find the sum of the series nC02 + nC12 + .. + nCk2