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Find $ a, b $ and $ c $ such that the following numbers are in AP: $ a, 7, b, 23, c $.
Given:
An A.P. $a,\ 7,\ b,\ 23,\ c$.
To do:
We have to find $a,\ b$ and $c$.
Solution:
$a,\ 7,\ b,\ 23,\ c$ are in AP
Therefore,
$7-a=d$ ......$( i)$
$b-7=d$ ....... $( ii)$
$23-b=d$ ....... $( iv)$
$c-23=d$ ....... $( v)$
From $( i)$ and ( ii)$
$7-a=b-7$
$\Rightarrow -b-a=-14$
$\Rightarrow a+b=14$ ....... $( vi)$
From $( ii)$ and $( iv)$
$b-7=23-b$
$\Rightarrow b+b=23+7$
$\Rightarrow 2b=30$
$\Rightarrow b=\frac{30}{2}$
$\Rightarrow b=15$, Put this in equation $( vi)$
$a+15=14$
$\Rightarrow a=14-15=-1$
Now, from $( iv)$ and $( v)$
$c-23=23-b$
$\Rightarrow c-23=23-15$
$\Rightarrow c-23=8$
$\Rightarrow c=8+23$
$\Rightarrow c=31$
Thus, the values of $a,\ b$ and $c$ are $-1,\ 15$ and $31$ respectively.
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