Discussion Forum

If a,b,c be positive integers such that b/a is an integer. If a,b,c are  in geometric  progression and the arithmetic mean of a,b,c is b+2, then the value of   $\frac{a^{b}+a-14}{a+1}$  is 

Answer:

Explanation:

Plan 

(i)  a,b,c are in G.P ,

 then they can be taken as a, ar, ar² where r,(r ≠ 0)  is the common ratio

 (ii) Arithmetic mean of x₁,x₂,............x_n

= $\frac{x_{1}+x_{2}+....+x_{n}}{n}$

  Let a,b,c are a,ar, ar²  , where rε N

Also,    $\frac{a+b+c}{3}=b+2$

$\Rightarrow$              a+ar+ar²= 3(ar)+6

$\Rightarrow$           ar²-2ar+a =6

$\Rightarrow$                    (r-1)² =  $\frac{6}{a}$

 $\because$             6/a must be perfect square   a ε N

  $\therefore$   a can be 6 only.

 $\Rightarrow$          $r-1=\pm1\Rightarrow r=2$

 and            $\frac{a^{b}+a-14}{a+1}=\frac{36+6-14}{7}=4$