Paragraph: The product of 5 consecutive terms of an AP is 229635....

Paragraph: The product of 5 consecutive terms of an AP is 229635. The first, second and fifth terms are in GP.
Question: What is the common difference?

Options

A $3$
B $4$
C $5$
D $6$

Correct answer

D. $6$

Step-by-step solution

Let the required terms of the given AP are-

\mathrm{a}-2 \mathrm{ d}, \mathrm{a}-\mathrm{d}, \mathrm{a}, \mathrm{a}+\mathrm{d}, \mathrm{a}+2 \mathrm{ d} .

According to the question,

\begin{aligned} & (a-2 d)(a-d) a(a+d)(a+2 d)=229635 \\ \Rightarrow & a\left(a^2-d^2\right)\left(a^2-4 d^2\right)=229635 ....(i)\\ \because & a-2 d, a-d, a+2 d \text{ are in G.P. } \\ \therefore & (a-d)^2=(a-2 d)(a+2 d) \\ \Rightarrow & (a-d)^2=a^2-4 d^2 \\ \Rightarrow & 5 d^2-2 a d=0 \Rightarrow d(5 d-2 a)=0 \Rightarrow 5 d=2 a \end{aligned}

....(ii) Solving equation (i) and (ii), we have,

\mathrm{d}=6 .

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