Chapter 11, Problem 1MCQ

To determine

To determine whether the below sequence is arithmetic, geometric, or neither:

  $5,-3,-12,-22,-\mathrm{33......}$

Answer to Problem 1MCQ

The given sequence is not arithmetic and geometric.

Explanation of Solution

Given:

  $5,-3,-12,-22,-\mathrm{33......}$

Concept Used:

A sequence is arithmetic if difference between it’s consecutive terms is same. An arithmetic sequence with first term a and common term d can be written in the form $a,a+d,a+2d,a+3d,\mathrm{......}$ .

A sequence is geometric if ratio of it’s consecutive terms is same. A geometric sequence with first term a and common ratio r can be written in the form $a,a\cdot r,a\cdot r^2,a\cdot r^3,\mathrm{......}$

Calculation:

In order to determine whether the below sequence is arithmetic, geometric, or neither:

  $5,-3,-12,-22,-\mathrm{33......}$

First, check whether this is arithmetic by checking whether it’s consecutive terms has common difference, as

  $\begin{array}{c}{a}_2-a_1=-3-5\\ =-8\\ a_3-a_2=-12-\left(-3\right)\\ -12+3\\ -9\end{array}$

Thus, each consecutive term of the given sequence does not have common difference thus the sequence is not arithmetic.

Now, check whether given sequence is a geometric check if there is a common ratio in it’s consecutive terms, and find ratio, as

  $\begin{array}{c}\frac{a_2}{a_1}=\frac{-3}{5}\\ \frac{a_3}{a_2}=\frac{-12}{-3}\\ =\frac{-3}{5}\ne \frac{-12}{-3}\\ \text{Thus,}\\ \frac{a_2}{a_1}\ne \frac{a_3}{a_2}\end{array}$

Thus, consecutive terms of given sequence doesn’t have common ratio thus the sequence is not geometric.

Thus, the given sequence is not arithmetic and geometric.