Question

Five numbers are in A.P. with common difference $\ne 0$ If the $1^{st},3^{rd}$ and $4^{th}$ terms are in G.P., then

• A. The $5^{th}$ term is always 0
• B. The $1^{st}$ term is always 0
• C. The middle term is always 0
• D. The middle term is always 2

Answer 1

Answer: (A)

The $5^{th}$ term is always 0

Let the numbers in A.P. be:

(a-2d), (a-d), a, (a+d), (a+2d)

Given, $d\ne 0$

1st, 3rd and 4th are in G.P.

$\therefore a^2=(a-2d)(a+d)$

$\Rightarrow a^2=a^2-2ad+ad-2d^2$

$\Rightarrow ad+2d^2=0$

$\Rightarrow d(a+2d)=0$

$\therefore (a+2d)=0$ [$d\ne 0$]

$\therefore$ 5th term = 0