Chapter 8.1, Problem 60E

To determine

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

Answer to Problem 60E

The first five terms are

  $25,20,15,10,5$ .

The pattern for nth term is $a_n=30-5n$ .

Explanation of Solution

Given information:

Given sequence

  $a_1=25,a_{k+1}=a_k-5$

Calculation:

Since the first term of the sequence is $a_1=25$ therefore, the subsequent terms can be computed from the recursive formula, $a_{k+1}=a_k-5$ as follows,

  $\begin{array}{c}{a}_2=a_1-5\\ =25-5\\ =20\\ a_3=a_2-5\\ =20-5\\ =15\\ a_4=a_3-5\\ =15-5\\ =10\\ a_5=a_4-5\\ =10-5\\ =5\end{array}$

Hence the first five terms are

  $25,20,15,10,5$ .

Now, the pattern for nth term can be obtained as

Consider the sequence $25,20,15,10,5$

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

The terms are $25,20,15,10,\mathrm{5...}{a}_n$

Rewrite the terms as shown below,

  $\begin{array}{c}25,20,15,10,\mathrm{5...}{a}_n=30-5,30-10,30-15,30-20,30-\mathrm{25...}{a}_n\\ =30-\left(5\times 1\right),30-\left(5\times 2\right),30-\left(5\times 3\right),30-\left(5\times 4\right),30-\left(5\times 5\right)\mathrm{...}{a}_n\end{array}$

Thus pattern followed is such that each term is 5 times $n$ subtracted from 30. Therefore , the general term is $a_n=30-5n$ .