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QuestionFind the first five terms of the sequence given by 2n+12^{n+1} for n=0,1,2,3,4n = 0, 1, 2, 3, 4. Determine the limit as nn approaches infinity.

Studdy Solution

STEP 1

Assumptions1. The sequence is defined by the formula n+1^{n+1} . The table provides the first five terms of the sequence3. We need to find the limit of the sequence as nn approaches infinity

STEP 2

First, let's write down the general term of the sequence using the given formula.
an=2n+1a_n =2^{n+1}

STEP 3

Now, we need to find the limit of the sequence as nn approaches infinity.
lim⁡n→∞an=lim⁡n→∞2n+1\lim{n \rightarrow \infty} a_{n} = \lim{n \rightarrow \infty}2^{n+1}

STEP 4

We know that for any positive constant kk, lim⁡n→∞kn=∞\lim{n \rightarrow \infty} k^n = \infty. Here, k=2k =2 is a positive constant. Therefore, the limit of the sequence as nn approaches infinity is infinity.
lim⁡n→∞2n+1=∞\lim{n \rightarrow \infty}2^{n+1} = \inftyThe correct choice is B. The limit does not exist or is ∞\infty or −∞-\infty.

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