Chapter 9.1, Problem 50E

To determine

To Verify: The nature of sign of all the other terms of a geometric sequence if the first two terms are positive.

Answer to Problem 50E

The given statement is always true.

Explanation of Solution

Given Information: The first two terms $a_1,a_2$ of a geometric sequenceare positive.

Concept Used: If the first term of a geometric sequence is $a$ , then the second term is $ar$ , where $r$ is the common ratio. Also, the $n^{th}$ term is

  $a_n=a{r}^{n-1}$

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

  $a_1=a,a_2=ar$

  $\Rightarrow a=a_1,r=\frac{a_2}{a_1}$

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

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

The expression $a_n=\frac{a_2^{n-1}}{a_1^{n-2}}$ is always positive because all the numbers involved in this expression, that is $a_1,a_2$ and $n$ ,are positive.