Chapter 9, Problem 52RE

a.

To determine

To find : The first term of the given arithmetic sequence.

Answer to Problem 52RE

The first term is 13.

Explanation of Solution

Given information :

The second and the sixth terms of an arithmetic sequence are 11.5 and 5.5 respectively.

Calculation :

The nth term of the arithmetic sequence is given by

  $a_n=a_1+\left(n-1\right)d$ …… (1)

Where $a_1$ is the first term and $d$ is the common difference.

Since, second term is 11.5. Therefore,

  $a_1+d=11.5$ …… (2)

And sixth term is 5.5. Therefore,

  $a_1+5d=5.5$ …… (3)

Subtract equation (1) from (2) and solve:

  $\begin{array}{l}{a}_1+5d-\left(a_1+d\right)=5.5-11.5\\ 4d=-6\\ d=\frac{-6}{4}\\ d=-1.5\end{array}$

Substitute $d=-1.5$ in equation (2) and solve:

  $\begin{array}{l}{a}_1+\left(-1.5\right)=11.5\\ a_1=11.5+1.5\\ a_1=13\end{array}$

Hence,

The first term is 13.

b.

To determine

To find : The common difference of the given arithmetic sequence.

Answer to Problem 52RE

The common difference is $-1.5$ .

Explanation of Solution

Given information :

The second and the sixth terms of an arithmetic sequence are 11.5 and 5.5 respectively.

Calculation :

The nth term of the arithmetic sequence is given by

  $a_n=a_1+\left(n-1\right)d$ …… (1)

Where $a_1$ is the first term and $d$ is the common difference.

Since, second term is 11.5. Therefore,

  $a_1+d=11.5$ …… (2)

And sixth term is 5.5. Therefore,

  $a_1+5d=5.5$ …… (3)

Subtract equation (1) from (2) and solve:

  $\begin{array}{l}{a}_1+5d-\left(a_1+d\right)=5.5-11.5\\ 4d=-6\\ d=\frac{-6}{4}\\ d=-1.5\end{array}$

Hence,

The common difference is $-1.5$ .

c.

To determine

To find : an explicit formula for the nth term.

Answer to Problem 52RE

Explicit formula for the nth term is $a_n=14.5-1.5n$ .

Explanation of Solution

Given information :

The second and the sixth terms of an arithmetic sequence are 11.5 and 5.5 respectively.

Calculation :

The nth term of the arithmetic sequence is given by

  $a_n=a_1+\left(n-1\right)d$ …… (1)

Where $a_1$ is the first term and $d$ is the common difference.

From (a) first term is $a_1=13$ and from (b) common difference is $d=-1.5$ substitute these values in equation (1) and solve:

  $\begin{array}{l}{a}_n=13+\left(n-1\right)\left(-1.5\right)\\ a_n=13-1.5n+1.5\\ a_n=14.5-1.5n\end{array}$

Hence,

Explicit formula for the nth term is $a_n=14.5-1.5n$ .