Chapter 12, Problem 9CT

To determine

Whether the sequence $\left\{-\frac{n}{2}+7\right\}$ is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric find the common ratio and sum of the first n terms.

Answer to Problem 9CT

Solution:

The sequence $\left\{-\frac{n}{2}+7\right\}$ is arithmetic with common difference $d=-\frac{1}{2}$ and the sum of first n terms is $\frac{n}{4}\left(27-n\right)$.

Explanation of Solution

Given information:

Sequence, $\left\{-\frac{n}{2}+7\right\}$.

Explanation:

Consider the sequence $\left\{-\frac{n}{2}+7\right\}$.

$a_n=-\frac{n}{2}+7$

$a_{n-1}=-\frac{(n-1)}{2}+7$

Check the difference between consecutive terms is constant or not.

$a_n-a_{n-1}=\left[-\frac{n}{2}+7\right]-\left[-\frac{(n-1)}{2}+7\right]$

$=-\frac{n}{2}+7+\frac{n-1}{2}-7$

$=-\frac{1}{2}$

The difference between consecutive terms is constant. Thus the sequence is arithmetic with common difference $d=-\frac{1}{2}$ and first term $a_1=-\frac{1}{2}+7=\frac{13}{2}$.

The sum of the first n terms is:

$S_n=\frac{n}{2}(a_1+a_n)$

$=\frac{n}{2}\left(\frac{13}{2}+\left(-\frac{n}{2}+7\right)\right)$

$=\frac{n}{2}\left(\frac{27}{2}-\frac{n}{2}\right)$

$=\frac{n}{4}\left(27-n\right)$

Thus the sequence $\left\{-\frac{n}{2}+7\right\}$ is arithmetic with common difference $d=-\frac{1}{2}$ and the sum of first n terms is $\frac{n}{4}\left(27-n\right)$.