Let the A.P. have first term $a$ and common difference $d$.

1. Write the required terms of the A.P.

The $n$th term of an A.P. is $T_n=a+(n-1)d$

So,

• $2^{\text{nd}}$ term: $T_2=a+d$
• $5^{\text{th}}$ term: $T_5=a+4d$
• $9^{\text{th}}$ term: $T_9=a+8d$

These are in G.P.

2. Use the condition for three numbers to be in G.P.

If $x,y,z$ are in G.P., then $y^2=xz$

Hence, $(a+4d)^2=(a+d)(a+8d)$

3. Expand both sides

Left side: $(a+4d)^2=a^2+8ad+16d^2$

Right side: $(a+d)(a+8d)=a^2+9ad+8d^2$

Equating: $a^2+8ad+16d^2=a^2+9ad+8d^2$

Cancel $a^2$: $8ad+16d^2=9ad+8d^2$

Rearrange: $8d^2-ad=0$ $d(8d-a)=0$

Since the A.P. is non-constant, $d\neq 0$. Therefore, $a=8d$

4. Find the three terms

Substitute $a=8d$:

• $T_2=a+d=8d+d=9d$
• $T_5=a+4d=8d+4d=12d$
• $T_9=a+8d=8d+8d=16d$

So the three terms are $9d,\;12d,\;16d$

5. Find the common ratio of the G.P.

$r=\frac{12d}{9d}=\frac{4}{3}$ Also, $\frac{16d}{12d}=\frac{4}{3}$

Thus the common ratio is $\boxed{\frac{4}{3}}$

6. Check options

• A: $1$ ❌
• B: $\frac{7}{4}$ ❌
• C: $\frac{8}{5}$ ❌
• D: $\frac{4}{3}$ ✅

Therefore, the correct option is D.