If a, b, c, d are in G.P, prove that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P.

We know that a, ar, ar², ar³,… are in G.P. with first term a & common ratio r.

Given a, b, c, d are in G.P.

So, a = a

b = ar

c = ar²

d = ar³

We want to show that

(aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in GP i.e to show common ratio are same

Now,

L.H.S

putting b = ar, c = ar²

R.H.S

putting c = ar², d = ar³, b = ar

Thus, L.H.S = R.H.S

Hence, (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in GP.