In Exercises 125–134, determine whether the sequence is monoto...

Question

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

Accepted Answer

Welcome back everyone. Determine whether the sequence c n equals 6 to the power of n + 1 + 4 power of n divided by 6 to the power of n converges where diverges. If it converges, find its limit. A converges to 3, B converges to 6. C converges to 0, and D diverges. So for this problem, let's go ahead and apply the nth term test of convergence of sequences. We want to evaluate the limit as n approaches infinity of cm. And in this case that be limit as n approaches infinity of 6 to the power of n + 1 + 4 n divided by 6 n. To solve this limit, we can simply divide to get the limit as n approaches infinity of 6 to the power of n plus 1 divided by 6 the power of n. Plus 4 to the power of n divided by 6 to the power of n. Let's go ahead and apply the properties of exponents so that the limit as n approaches infinity of 6 multiplied by 6 to the power of n divided by 6 to the power of n for the first term, which basically allows us to cancel out 6 to the power of n plus 4 divided by 6 raise to the power of n. Now let's simplify slightly. This is limit as n approaches infinity of 6 + 2/3 raise to the power of n. And now we can split it into two limits. Limits and approaches infinity of 6 plus. Limit is n approaches infinity of 2/3 to the power of n. So the first limit is 6 because this is the limit of a constant, and the second limit is 0 because as n approaches infinity. Our base, which is between 1 and 1, right, because 2/3 is between -1 and 1. Specifically, when we raise a number within that range to an infinitely large number, the limit is essentially 0 since we have geometric series with a common ratio. Absolute value of r less than 1. So we get 6 plus 0, which is 6. Now what can we conclude? Well, Since the limit is a finite constant. The sequence converges and it converges towards the value of that limit, so towards 6. So the answer to this problem is option B converges to 6. Thank you for watching.

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

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In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.
aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ