Q. 51 - Calculus

Question

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

$\left\{k^2-5k\right\}$

Step-by-Step Solution

Answer

The given sequence is strictly increasing for $k\ge 2.$

Step 1. Given Information.

The given sequence is $\left\{k^2-5k\right\}.$

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 is $a_k=k^2-5k.$

So, the term $a_{k+1}$ is

  $$\begin{gathered} a_{k+1}={\left(k+1\right)}^2-5\left(k+1\right) \\ {a}_{k+1}=k^2+2k+1-5k-5 \\ {a}_{k+1}=k^2-3k-4 \end{gathered}$$

According to the test,

$$\begin{gathered} a_{k+1}-a_k\ge 0 \\ {k}^2-3k-4-\left(k^2-5k\right)\ge 0 \\ {k}^2-3k-4-k^2+5k\ge 0 \\ 2k-4\ge 0 \\ k\ge 2 \end{gathered}$$

Thus, the sequence is strictly increasing for $k\ge 2.$