Question

Consider an increasing sequence of seven positive numbers $a_1,a_2,\dots \dots ,a_7$ such that $a_1,a_2,a_3$ are in A.P. with common difference d $a_3,a_4,a_5$ are in A.P. with common difference $2d$ $a_5,a_6,a_7$ are in G.P. with common ratio $r$ if $a_5-a_1=12,a_6=7a_3$ and $a_7=63a_1$, then

• A. d equals 2
• B. r equals 3
• C. $a_1+r=5$
• D. r equals $\frac{1}{3}$

Answer 1

Answer: (A), (B), (C)

$a_3=a_1+2d,a_5=a_1+2d+2(2d)=a_1+6d$
$\Rightarrow a_5-a_1=6d=12\Rightarrow d=2$
$a_6=\left(a_1+12\right)r=7\left(a_1+4\right)$
$\left(a_1+12\right)r^2=63a_1$
(i) $÷$ (ii) $\Rightarrow r=\frac{9a_1}{a_1+4}$ put in equation (i)
$\therefore 9a_1^2+108a_1=7a_1^2+112+56a_1$
$2a_1^2+52a_1-112=0$
$a_1^2+26a_1-56=0$
$a_1^2+28a_1-2a_1-56=0$
$a_1=2,r=3.$