Question

if a,b,(c+1) are in G.P. and a=(b-c)^2 then a,b,c are in​

Answer 1

If a,b,(c+1) are in G.P. and a=(b-c)^2 then a,b,c are in​ Arithmetic progression.

Given, a, b, c+1 are in G. P

So,

b^2 = a ( c + 1)

b^2 = ac + a

a = b^2 - ac

Also,

a =( b-c) ^2

a = b^2 + c^2 - 2bc

From both equations,

b^2 - ac = b^2 + c^2 - 2bc

-ac +2bc = c²

-a +2b = c

2b = a + c.

So, a, b, c are in Arithmetic progression.

Conceptual knowledge :

For x, y, z to be in

• Arithmetic progression : 2y = x + z

• Geometric progression : y^2 = xz

• Harmonic progression, 1/x, 1/y, 1/z should be in A. P.