Chapter 9.1, Problem 49E

To determine

To Verify: The nature of sign of all the other terms of an arithmetic sequence if the first two terms are negative.

Answer to Problem 49E

The given statement is false.

Explanation of Solution

Given Information: The first two terms $a_1,a_2$ of an arithmetic sequence are negative.

Concept Used: If the first term of an arithmetic sequence is $a$ , then the second term is $a+d$ , where $d$ is the common difference. Also, the $n^{th}$ term is

  $a_n=a+\left(n-1\right)d$

Let the first term be $a_1$ and the second term be $a_2$ , Then

  $a_1=a,a_2=a+d$

  $\Rightarrow a=a_1,d=a_2-a_1$

Substitute the expressions for $a,d$ calculated above in terms of $a_1,a_2$ in the expression for $n^{th}$ term to write $a_n$ as

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ =a_1+\left(n-1\right)\left(a_2-a_1\right)\\ =n\left(a_2-a_1\right)+2a_1-a_2\end{array}$

If $a_2>a_1$ , the expression $a_n=n\left(a_2-a_1\right)+2a_2-a_1$ can be positive for a sufficiently large value of $n$ . This is because if $a_2>a_1$ , then the coefficient of $n$ in the expressionwill be positive.

Therefore, the given statement isnot correctif the second term is greater than the first term, even though if both of them are negative.