Chapter 12, Problem 8CT

To determine

Whether the sequence $-2,-10,-18,-26,\mathrm{...}$ 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 8CT

Solution:

The sequence $-2,-10,-18,-26,\mathrm{...}$ is arithmetic with common difference $d=-8$ and the sum of first n terms is $n(2-4n)$.

Explanation of Solution

Given information:

Sequence, $-2,-10,-18,-26,\mathrm{...}$

Explanation:

Consider the sequence $-2,-10,-18,-26,\mathrm{...}$

Check the difference between consecutive terms is constant or not.

$-10-(-2)=-8,-18-(-10)=-8$

The difference between consecutive terms is constant. Thus the sequence is arithmetic with common difference $d=-8$ and first term $a_1=-2$.

Calculate n^th term of the sequence:

$a_n=a_1+(n-1)d$

$=-2+(n-1)(-8)$

$=-2+-8n+8$

$=6-8n$

The sum of the first n terms is:

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

$=\frac{n}{2}\left(-2+6-8n\right)$

$=\frac{n}{2}\left(4-8n\right)$

$=n\left(2-4n\right)$

Thus the sequence $-2,-10,-18,-26,\mathrm{...}$ is arithmetic with common difference $d=-8$ and the sum of first n terms is $n\left(2-4n\right)$.