Question

The sixth term of a GP is -2/27 and its first term is 18.what is the common ratio

Answer 1

In a geometric progression (GP), the \( n \)-th term can be expressed as:

\[
a_n = a \cdot r^{n-1}
\]

where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

Given:
- The first term \( a = 18 \),
- The sixth term \( a_6 = -\frac{2}{27} \).

Using the formula for the sixth term:

\[
a_6 = a \cdot r^{6-1} = a \cdot r^5
\]

Substituting the values:

\[
-\frac{2}{27} = 18 \cdot r^5
\]

To isolate \( r^5 \), divide both sides by 18:

\[
r^5 = \frac{-\frac{2}{27}}{18}
\]

Calculating the right-hand side:

\[
r^5 = -\frac{2}{27 \cdot 18} = -\frac{2}{486} = -\frac{1}{243}
\]

Now, we want to find \( r \). To do this, we take the fifth root of both sides:

\[
r = \sqrt[5]{-\frac{1}{243}}
\]

Since \( 243 = 3^5 \), we can express \( r \) as:

\[
r = \sqrt[5]{-\frac{1}{3^5}} = -\frac{1}{3}
\]

Therefore, the common ratio \( r \) is:

\[
\boxed{-\frac{1}{3}}
\]