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Numerade Educator



Problem 65 Hard Difficulty

If $ \$ $1000 is invested at $ 6 \% $ interest, compounded annually, then after $ n $ years the investment is worth $ a_n = 1000(1.06)^n $ dollars.
(a) Find the first five terms of the sequence $ \{ a_n\}. $
(b) Is the sequence convergent or divergent? Explain.


a. 1338.23
b. diverges


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Video Transcript

for part A. Let's go ahead and find the first five terms. So a one using the formula here just plug in and equals one. That's just 1060.0 dollars, of course. So this time 1.6 squared going to the calculator And for this one, a three 1.6 cube going to the calculator again, a four coming up next and go to the calculator for this one. And finally a five plug in and equals five in the formula. Then go to the calculator for this. So this takes care of part A and it looks like it will diverge. But let's show this more rigorously so for part B. Now I claim that this will diverge to infinity. Since the following Excuse me. This should have been in here. Now this is infinity. Now I'll verify this in this equation here and if we just multiply by 1000 well, that doesn't change the limit. So to show this, what you can do is rewrite this limit first. Maybe let's use X instead of n so we could use low petals rule. So here are right. This is Now, this fact is just coming from rewriting any number. Why, as e to the l N y. And using your log properties. So this becomes e infinity, Ellen, 1.6 This number over here is positive. So this is just either the infinity equals infinity. So this shows that this limits infinity. And if we multiply by 1000 this shows that the limit and goes to infinity of a n equals infinity. So the sequence a and diverges and that's our final answer.