1. Tardigrade
  2. Question
  3. Mathematics
  4. Consider an increasing sequence of seven positive numbers a1, a2, ldots ldots, a7 such that a1, a2, a3 are in A.P. with common difference d a3, a4, a5 are in A.P. with common difference 2 d a5, a6, a7 are in G.P. with common ratio r if a5-a1=12, a6=7 a3 and a7=63 a1, then

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

Sequences and Series

Solution:

$a_3=a_1+2 d, \quad a_5=a_1+2 d+2(2 d)=a_1+6 d $
$\Rightarrow a_5-a_1=6 d=12 \Rightarrow d=2 $
$a_6=\left(a_1+12\right) r=7\left(a_1+4\right) $
$\left(a_1+12\right) r^2=63 a_1$
(i) $\div$ (ii) $\Rightarrow r=\frac{9 a_1}{a_1+4}$ put in equation (i)
$\therefore \quad 9 a _1^2+108 a _1=7 a _1^2+112+56 a _1 $
$2 a _1^2+52 a _1-112=0$
$a_1^2+26 a_1-56=0 $
$a_1^2+28 a_1-2 a_1-56=0 $
$a_1=2, r=3 .$