If the 2nd term of a G.P is $\frac{8}{9}$ and the 6th term is 4$\frac{1}{2}$. Find the common ratio.

1. Analyze the Question

We are given two terms of a Geometric Progression (G.P.):

• The 2nd term ($ar$) = $\frac{8}{9}$
• The 6th term ($ar^5$) = $4\frac{1}{2}$ (which is $\frac{9}{2}$ as an improper fraction) We need to find the common ratio ($r$).

2. Explain the Concept

In a G.P., the $n^{th}$ term is given by the formula:

where:

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

By dividing the expression for the 6th term by the expression for the 2nd term, we can eliminate $a$ and solve for $r$.

3. Show the Solution

Step 1: Write down the equations based on the formula.

• $T_2 = ar^{2-1} = ar = \frac{8}{9}$ --- (Equation 1)
• $T_6 = ar^{6-1} = ar^5 = \frac{9}{2}$ --- (Equation 2)

Step 2: Divide Equation 2 by Equation 1.

The $a$ cancels out:

Step 3: Solve for $r$. To find $r$, take the fourth root of both sides:

Since $3^4 = 81$ and $2^4 = 16$:

4. Identify the Answer

Comparing our result to the options provided: A. 2 B. $\frac{3}{2}$ C. $\frac{2}{3}$ D. 3

The correct option is B.

5. Tip

When dealing with G.P. problems involving two terms, always remember the "gap" rule: the ratio of the $n^{th}$ term to the $m^{th}$ term is $r^{(n-m)}$. Here, the gap between the 6th and 2nd term is $6 - 2 = 4$, so $r^4 = \frac{T_6}{T_2}$. This shortcut saves you from having to solve for the first term ($a$) separately!