Question

If the 2$^{\text{nd}}$, 5$^{\text{th}}$ and 9$^{\text{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is

• A. $\dfrac{8}{5}$
• B. $\dfrac{4}{3}$
• C. $1$
• D. $\dfrac{7}{4}$

Hint:

If three quantities are in G.P., then the square of the middle term equals the product of the first and third.

Answer 1

The Correct Option is B

Step 1: Let the A.P. have first term $a$ and common difference $d$. \[ \text{2nd term} = a+d,\quad \text{5th term} = a+4d,\quad \text{9th term} = a+8d \]
Step 2: Since these three terms are in G.P., the square of the middle term equals the product of the other two: \[ (a+4d)^2 = (a+d)(a+8d) \]
Step 3: Expand both sides: \[ a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2 \]
Step 4: Simplify: \[ ad - 8d^2 = 0 \] \[ d(a-8d)=0 \]
Step 5: Since the A.P. is non-constant, $d \neq 0$, hence: \[ a = 8d \]
Step 6: Substitute $a=8d$ in the terms: \[ \text{2nd term} = 9d,\quad \text{5th term} = 12d,\quad \text{9th term} = 16d \]
Step 7: The common ratio of the G.P. is: \[ r=\frac{12d}{9d}=\frac{4}{3} \]