The second and fifth terms of a G.P are 1 and \(\frac{1}{8}\) respectively. Find the common ratio

A geometric progression (G.P.) is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. If the first term is \(a\) and the common ratio is \(r\), then the \(n\)th term is:

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

According to the problem:

• The second term is \(1\). So, \[ a_2 = a \cdot r^{2-1} = a \cdot r = 1 \]
• The fifth term is \(\frac{1}{8}\). So, \[ a_5 = a \cdot r^{5-1} = a \cdot r^4 = \frac{1}{8} \]

From the first equation, \[ a \cdot r = 1 \implies a = \frac{1}{r} \]

Substitute \(a = \frac{1}{r}\) into the equation for the fifth term: \[ a_5 = \frac{1}{r} \cdot r^4 = r^{4-1} = r^3 \] So, \[ r^3 = \frac{1}{8} \]

To solve for \(r\), take the cube root of both sides: \[ r = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \]

This means the common ratio is \(\frac{1}{2}\).

• \(\frac{1}{2}\) is correct because raising it to the third power gives \(\frac{1}{8}\), and it satisfies all the given terms in the sequence.

Summary Table:

This confirms the value of \(r\): \(\frac{1}{2}\) is the common ratio.