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.

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