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

Analysis

To solve this problem, we need to use the formula for the $n^{th}$ term of a Geometric Progression (G.P.):

Where:

• $a$ is the first term
• $r$ is the common ratio
• $n$ is the position of the term

Concept

In a G.P., each term is obtained by multiplying the previous term by a constant value called the common ratio ($r$). By setting up equations for the second and fifth terms, we can create a system of equations to isolate $r$.

Step-by-Step Solution

Step 1: Write the equations for the given terms.

• For the second term ($n=2$): $$T_2 = ar^{2-1} = ar = 1 \quad \text{--- (Equation 1)}$$
• For the fifth term ($n=5$): $$T_5 = ar^{5-1} = ar^4 = \frac{1}{8} \quad \text{--- (Equation 2)}$$

Step 2: Divide Equation 2 by Equation 1 to eliminate $a$.

Step 3: Solve for $r$ by taking the cube root of both sides.

Since $2 \times 2 \times 2 = 8$, the cube root of $\frac{1}{8}$ is $\frac{1}{2}$.

Conclusion

The common ratio of the G.P. is $\frac{1}{2}$.

Exam Tip

When dealing with G.P. problems where you are given two terms, always remember that the ratio of the two terms $T_n$ and $T_m$ is given by $\frac{T_n}{T_m} = r^{n-m}$. In this case:

This shortcut saves you time during the JAMB exam!