Chapter 11.7, Problem 2MRE

To determine

To calculate:

Tell whether the sequence $\frac{9}{16},\frac{3}{4},\mathrm{1..}$ is arithmetic, geometric or neither and find the formula for the nth term of the sequence.

Answer to Problem 2MRE

The given sequence $\frac{9}{16},\frac{3}{4},\mathrm{1..}$ is a geometric sequence and the nth term of the sequence is. $a_n=\frac{9}{16}\cdot \frac{3^{n-1}}{4}$

Explanation of Solution

Given information:

The given sequence is,

  $\frac{9}{16},\frac{3}{4},\mathrm{1..}$

Formula:

The formula of geometric sequence is,

  $a_n=a_1{r}^{n-1}$

Where,

  $r$ : Common ratio value

  $a_n$ : Nth term

  $a_1$ : $1st$ Term of the geometric sequence

Calculation:

The given sequence is,

  $\frac{9}{16},\frac{3}{4},\mathrm{1..}$

Here, check if the sequence is arithmetic

The difference between the two consecutive terms is not same. Thus the sequence is not arithmetic.

Now, check if the sequence is geometric.

  $\frac{9}{16}/\frac{3}{4}=\frac{3}{4}$

The common ratio exists in the given sequence. Thus the sequence is geometric sequence.

Now, to find the nth term of the geometric sequence, $a_n=a_1\cdot r^{n-1}$

Here in the given sequence,

  $a_1=\frac{9}{16}$

  $r=\frac{3}{4}$

Substituting all the values in the given formula,

  $a_n=\frac{9}{16}\cdot \frac{3^{n-1}}{4}$

The nth term of the geometric sequence is $a_n=\frac{9}{16}\cdot \frac{3^{n-1}}{4}$