Chapter 4.7, Problem 51PPE

To determine

To find: The second, fourth and eleventh terms of the sequence using explicit formula.

Answer to Problem 51PPE

The $2nd,4th\text{}\text{ and }11th$ terms of the given arithmetic sequence are

  $A\left(2\right)=-2,A\left(4\right)=8\text{ and }A\left(11\right)=43.$ .

Explanation of Solution

Given:

  $A\left(n\right)=-7+\left(n-1\right)\left(5\right)$ .

Concept used:

Explicit formula arithmetic series:

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)d$ .

Where $d$ is common difference.

  $A\left(1\right)$ is first term.

Calculation:

Consider that the rule for finding nth term of an arithmetic sequence is given by equation $\left(i\right)$ .

Step $1$ : recall that the nth term of an arithmetic sequence with the first term $A\left(1\right)$ and common difference $d$ is given by:

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)d\mathrm{........}\left(ii\right)$

So, the first term of the given sequence is $-7$ and the common difference is $5$ . Now to find the second, fourth and eleventh terms of the sequence substitute $n=2,4\text{ and }11$ in equation $\left(i\right)$ a dn simplify to get:

  $$ .

Find the second, fourth and eleventh terms of the sequence described by the rule

  $\begin{array}{l}A\left(2\right)=-7+\left(2-1\right)\left(5\right).\\ \Rightarrow -2.\\ A\left(4\right)=-7+\left(4-1\right).\left(5\right).\\ \Rightarrow 8.\end{array}$

The eleventh term of the sequence is given by:

  $\begin{array}{l}A\left(11\right)=-7+\left(11-1\right)\left(5\right).\\ \Rightarrow -7+10\left(5\right).\\ \Rightarrow 43.\end{array}$

Hence, the $2nd,4th\text{}\text{ and }11th$ terms of the given arithmetic sequence are

  $A\left(2\right)=-2,A\left(4\right)=8\text{ and }A\left(11\right)=43.$