Question

08. If a1, a2, a3 represents first, second and third terms of an AP respectively, the first
term is 2 and (a1 + a2) a3 is minimum, then the common difference is equal to
(a) 5/2 (b) - 5/2 (C) 2/5 (d) - 2/5
​

Answer 1

Answer:

The required value of d is $\frac{-5}{2}$

option (b) is correct

Step-by-step explanation:

Given: $a_1=2$

Let d be the common differernce

Then

$a_2=2+d$

$a_3=2+2d$

$(a_1+a_2)a_3$

$=(2+2+d)(2+2d)$

$=(4+d)(2+2d)$

$=8+8d+2d+2d^2$

$=8+10d+2d^2$

$=2d^2+10d+8=y(say)$

$y=2d^2+10d+8$

$\frac{dy}{dd}=4d+10$

$\frac{d^2y}{dd^2}=4$

Now,

$\frac{dy}{dd}=0$

$4d+10=0$

$2d+5=0$

$d=\frac{-5}{2}$

when $d=\frac{-5}{2}$,

$\frac{d^2y}{dd^2}=4\:>\:0$

By second derivative test, y attains minimum at $d=\frac{-5}{2}$