Chapter 8.1, Problem 59E

To determine

To write: the first five terms of the sequence defined recursively and write the pattern for nth term.

Answer to Problem 59E

The first five terms are

  $6,8,10,12,14$ .

The pattern for nth term is $a_n=2n+4$ .

Explanation of Solution

Given information:

Given sequence

  $a_1=6,a_{k+1}=a_k+2$

Calculation:

Since the first term of the sequence is $a_1=6$ therefore, the subsequent terms can be computed from the recursive formula, $a_{k+1}=a_k+2$ as follows,

  $\begin{array}{c}{a}_2=a_1+2\\ =6+2\\ =8\\ a_3=a_2+2\\ =8+2\\ =10\\ a_4=a_3+3\\ =10+2\\ =12\\ a_5=a_4+2\\ =12+2\\ =14\end{array}$

Hence the first five terms are

  $6,8,10,12,14$ .

Now, the pattern for nth term can be obtained as

Consider the sequence $6,8,10,12,\mathrm{14...}$

For $n=1,2,3,4,5,\mathrm{...}n$

The terms are $6,8,10,12,\mathrm{14...}{a}_n$

Rewrite the terms as shown below,

  $\begin{array}{c}6,8,10,12,\mathrm{14...}{a}_n=2+4,4+4,6+4,8+4,10+\mathrm{4...}{a}_n\\ =\left(2\times 1\right)+4,\left(2\times 2\right)+4,\left(2\times 3\right)+4,\left(2\times 4\right)+2,\left(2\times 5\right)+\mathrm{4...}{a}_n\end{array}$

Thus pattern followed is such that each term is 4 more than 2 times $n$ . Therefore , the general term is $a_n=2n+4$ .