Three non-zero numbers `a ,b ,a n dc` are in A.P. Increasing `a` by 1 or increasing `c` by 2, the numbers are in G.P. Then find `bdot`

Explanation

a,b, and c are in A.P. Hence,
2b=a+c (1)
Again by the given condition, a+1,b,and c are in G.P. and a,b, and c+2 are in G.P. Hence,
`b^(2)=(a+1)c` (2)
and `b^(2)=a(c+2)` (3)
By (2) and (3),
(a+1)c=a(c+2)
or ac+c=ac+2a
or c=2a
Equation (2) gives `b^(2)=(a+1)2a`
Also,Eq.(1) gives
`2b=a+2a=3a`
or `b=(3a)/2`
or `(9a^(2))/4=(a+1)2a`
or `(9a)/8=a+1`
or a=8
or c=2(8)=16
`rArr` 2b=8+16=24
or b=12