If ( a, 7, b, 23, c ) form a finite ( A P ), find the values of ( a, b ) and ( c ).

Given:

\( a, 7, b, 23, c \) form a finite \( A P \).

To do: 

We have to find $a,\ b$ and $c$.

Solution:

$a,\ 7,\ b,\ 23,\ c$ are in AP

Therefore,

$7-a=d$ ......$(i)$

$b-7=d$   ....... $( ii)$

$23-b=d$  ....... $( iii)$

$c-23=d$  ....... $( iv)$

From $( i)$ and ( ii)$

$7-a=b-7$

$\Rightarrow -b-a=-14$

$\Rightarrow a+b=14$ ....... $( v)$

From $( ii)$ and $( iii)$

$b-7=23-b$

$\Rightarrow b+b=23+7$

$\Rightarrow 2b=30$

$\Rightarrow b=\frac{30}{2}$

$\Rightarrow b=15$, Put this in equation $( v)$

$a+15=14$

$\Rightarrow a=14-15=-1$

Now, from $( iii)$ and $( iv)$

$c-23=23-b$

$\Rightarrow c-23=23-15$

$\Rightarrow c-23=8$

$\Rightarrow c=8+23$

$\Rightarrow c=31$

Thus, the values of $a,\ b$ and $c$ are $-1,\ 15$ and $31$ respectively.