Question

If $1000 is invested at 8% interest, compounded annually, then after n years the investment is worthan = 1000(1.08)ndollars.(a) Find the first five terms of the sequence {an}. (Round your answers to the nearest cent.)a1 = $a2 = $a3 = $a4 = $a5 = $(b) Is the sequence convergent or divergent?

Answer 1

Diverge

Explanation:

(a)

1st year:
`1000 * 1.08^1 = \$1080`

2nd year:
`1000 * 1.08^2 = \$1166.4`

3rd year:
`1000 * 1.08^3 = \$1259.71`

4th year:
`1000 * 1.08^4 = \$1360.49`

5th year:
`1000 * 1.08^5 = \$1469.33`

(b) The sequence is divergent, because if we take the derivative of the function with respect to n year:

`(1000*1.08^n)^(') = 1000ln(n)*1.08^n`

This is a positive, meaning the slope of the function is positive. If we take the second derivative using product rule

`(1000ln(n)*1.08^n)^(') = 1000(ln(n))/(n)1.08^n`

This is also positive when n > 0. Therefore, the slope is positive and increasing. This means the sequence diverges.