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

Answer:

Diverge

Step-by-step 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

$(1000ln(n)*1.08^n)^{'} = 1000\frac{ln(n)}{n}1.08^n$

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