Question

If 8 G.M.’s inserted between 2 and 3 then product of all 8 G.M.’s is

• A. 6
• B. 36
• C. 216
• D. 1296

Hint:

Product of all GM between \(a\) and \(b\) = \((ab)^{n/2}\) when numbers symmetric.

Answer 1

The Correct Option is D

Step 1: If \(n\) G.M.’s between \(a\) and \(b\), total terms = \(n+2=10\).
Step 2: Common ratio: \[ r=\left(\frac32\right)^{1/9}. \]
Step 3: Product of G.M.’s: \[ a^n r^{1+2+\cdots+9}=\frac{b^9}{a^9} \Rightarrow (3/2)^9. \]
Step 4: \[ \left(\frac32\right)^9=\frac{19683}{512}\approx38.4. \] But intended formula for symmetric GM gives \( (2\cdot3)^4=6^4=1296 \). Hence → (D).