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In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

kk+2

Short Answer

Expert verified

The given sequence is strictly increasing fork≥1.

Step by step solution

01

Step 1. Given Information.

The given sequence iskk+2.

02

Step 2. Use the difference test. 

To analyze the monotonicity of the given sequence we will use the difference test.

Let the general term of the sequence isak=kk+2.

So, the term ak+1is

ak+1=k+1k+1+2.

According to the test,

ak+1-ak=k+1k+3-kk+2=k+1k+2-kk+3k+3k+2=k2+k+2k+2-k2-3kk+3k+2=2k+3k+2

Now,role="math" localid="1649230532100" ak+1-ak≥0forallk≥1.

Thus, the sequence is strictly increasing fork≥1.

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