Question

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

The correct option is A the $5^{th}$ term is always $0$

Let the line number is in AP are
$(a-2d),(a-d),a,(a+d),(a+2d)$
where $d\ne 0$
Given $1^{st},3^{rd}$ and $4^{th}$ terms are in GP
$\Rightarrow$ $a^2=(a-2d)(a+d)$
$\Rightarrow$ $a^2=a^2-2ad+ad-2d^2$
$\Rightarrow$ $2d^2+ad=0$ $\Rightarrow$ $d(2d+a)=0$
$\because$ $d\ne 0$
$\therefore$ $a+2d=0$ $\Rightarrow$ $a=-2d$
Hence terms are
$-4d,-3d,-2d,-d,0$
$\therefore$ The fifth term is always $0$