Question

Five numbers are in $A.P.$ with common difference $≠ \, 0$ 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
$(a-2 \,d),(a-d), a,(a+d),(a+2 \,d)$
where, $d \neq 0$
Given, 1 st, 3rd and 4th terms are in GP.
$\Rightarrow a^{2}=(a-2\, d)(a+d)$
$\Rightarrow a^{2}=a^{2}-2\, a d+a d-2\, d^{2}$
$\Rightarrow 2 d^{2}+a d=0$
$ \Rightarrow d(2 \,d+a)=0$
$\because d \neq 0$
$\therefore a+2\, d=0$
$ \Rightarrow a=-2 \,d$
Hence, terms are
$-4 \,d,-3 \,d,-2 d,-d, 0$
$\therefore $ The fifth term is always 0 .