Chapter 8.1, Problem 49E

To determine

To check : First two terms of an arithmetic sequence are negative, then all its terms are negative or not

Answer to Problem 49E

False.

Explanation of Solution

Given information :

The first two terms of an arithmetic sequence are negative.

Take an example of an arithmetic sequence:

  $\begin{array}{c}{a}_n=-5+2\left(n-1\right)\\ \text{For},n=1\\ a_1=-5+2\left(1-1\right)\\ =-5\\ \text{For,}n=2\\ a_2=-5+2\left(2-1\right)\\ =-3\\ \text{For},n=3\\ a_3=-5+2\left(3-1\right)\\ =-1\\ \text{But for,}n=4\\ a_4=-5+2\left(4-1\right)\\ =1\end{array}$

Here, for $n=4$ the arithmetic sequence $a_n=-5+2\left(n-1\right)$ is non-negative.

Hence, if first two terms of an arithmetic sequence are negative, then all its terms are not necessary negative.