Chapter 8.3, Problem 11E

To determine

To check: Whether the sequence is geometric.

Answer to Problem 11E

Yes, the given sequence is geometric.

Explanation of Solution

Given information:

The sequence:

  $\frac{1}{2},\frac{1}{6},\frac{1}{18},\frac{1}{54},\frac{1}{162},\mathrm{......}$

Formula used:

The geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Check for GP:

  $\frac{a_2}{a_1}=\frac{a_3}{a_2}$

Where $a_1,a_2,a_3$ are first, second and third term of GP respectively.

  $\frac{1}{2},\frac{1}{6},\frac{1}{18},\frac{1}{54},\frac{1}{162},\mathrm{......}$

Recall that Check for GP is:

  $\begin{array}{l}\frac{a_2}{a_1}=\frac{a_3}{a_2}\\ \\ a_1=\frac{1}{2}\\ a_2=\frac{1}{6}\\ a_3=\frac{1}{18}\end{array}$

  $\begin{array}{l}\frac{\frac{1}{6}}{\frac{1}{2}}=\frac{\frac{1}{18}}{\frac{1}{6}}\\ \\ \frac{2}{6}=\frac{6}{18}\\ \\ \frac{1}{3}=\frac{1}{3}\\ \\ LHS=RHS\end{array}$

the ratio of second term to first term is equal to the ratio of third term to second term.

Yes, the given sequence is geometric.