about to only mathematics

Explanation

(i) If a, b, c are in GP, then they can be taken as `a, ar, ar^(2)` where, `(r !=0)` is the common ratio
(ii) Arithmetic mean of `x_(1), x_(2),.., x_(n) = (x_(1) + x_(2) + ....+ x_(n))/(n)`
Let a, b, c be a, `ar, ar^(2)`, where `r in N`
Also, `(a + b +c)/(3) = b + 2`
`rArr a + ar + ar^(2) = 3 (ar) + 6`
`rArr ar^(2) - 2ar + a = 6`
`rArr (r -1)^(2) = (6)/(a)`
since, 6/a must be perfect square and `a in N`
So, a can be 6 only
`rArr r -1 = +- 1 rArr r = 2`
and `(a^(2) + a -14)/(a + 1) = (36 + 6 - 14)/(7) = 4`