Chapter 8.3, Problem 121E

To determine

To explain: the use of first two terms of a geometric sequence and to find the $n$ th term.

Answer to Problem 121E

The $n$ th term of a geometric sequence by using the first terms is given by

  $a_n=\left(a_1{}^{2-n}\right)\left(a_2{}^{n-1}\right)$

Explanation of Solution

Given information:

Calculation:

Let us consider a geometric sequence $a_1,a_1{r}^1,a_1{r}^2,a_1{r}^3,\mathrm{...}{a}_1{r}^{n-1},\mathrm{...}$

Not determine the $n$ th term of geometric sequence by using the first two terms as shown below:

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

Thus we have to divide the second term to obtain the common ratio. The $n$ th term of a geometric sequence is obtained by multiplying the first term and the common ratio raised to the $\left(n-1\right)$ th power.

The $n$ th term of a geometric sequence by using the first terms is given by

  $a_n=\left(a_1{}^{2-n}\right)\left(a_2{}^{n-1}\right)$