Chapter 9.1, Problem 52E

To determine

To Find:The $1^{st}$ term of a geometric sequence, two of whose other terms are given.

Answer to Problem 52E

The first term of the sequence is $5$ .

Explanation of Solution

Given Information: The $2^{nd}$ term and the $3^{nd}$ term of the geometric sequence are $2.5$ and $1.25$ respectively.

Concept Used:

• The $n^{th}$ term of a geometric series with first term $a$ and common ratio $r$ is given by
$a_n=a{r}^{n-1}$

Calculation: The second term is $a_2=2.5$ . So

  $a_2=a{r}^{2-1}=2.5$

  $\Rightarrow ar=2.5$ (1)

The third term is $1.25$ . So

  $a_3=a{r}^{3-1}=1.25$

  $\Rightarrow a{r}^2=1.25$(1)

Now, divide equation (1) by equation (2)to write

  $\frac{a{r}^2}{a{r}^1}=\frac{1.25}{2.5}$

  $\Rightarrow r=\frac{1}{2}$

Now, put the calculated value $r=\frac{1}{2}$ in equation (1) to write

  $a\times \frac{1}{2}=2.5$

  $\Rightarrow a=5$

Therefore, the first term of the sequence is $5$ .