Chapter 12, Problem 8CT

To determine

To find: The common ratio and the sum of the first $n$ terms.

Answer to Problem 8CT

The sum of the sequence is $n\left(2-4n\right)$ .

Explanation of Solution

Given information:

The given sequence is $-2,-10,-18,-\mathrm{26.................}$ .

Calculation:

Calculate the difference between the first and the second terms.

  $-10-\left(-2\right)=-8$

Calculate the difference between the second and the third terms.

  $-10-\left(-10\right)=-8$

Since the difference of the successive terms is not constant, we can say that the given sequence with common difference is $-8$ .

The sum $s_n$ of the first $n$ terms of an arithmetic sequence $\left\{a_n\right\}$ with the first term $a_1$ and common difference is $d$ .

Formula:

  $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Substitute the values in formula,

  $\begin{array}{c}{S}_n=\frac{n}{2}\left[2\left(-2\right)+\left(n-1\right)\left(-8\right)\right]\\ =\frac{n}{2}\left(4-8n\right)\\ =n\left(2-4n\right)\end{array}$

Therefore, the sum of the sequence is $n\left(2-4n\right)$ .