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)

Let the tine number is in AP are
$(a-2d),(a-d),a,(a+d),(a+2d)$
where, $d\ne 0$
Given, 1 st, 3rd and 4th 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 .