# If a, b, c are in A.P.; b, c, d are in G.P. and are in A.P. prove that a, c, e are in G.P.

It is given that a, b, c are in AP

b = (a + c)/2 …(1)

Also given that b, c, d are in GP

c2 = bd …(2)

Also,

are in AP

So, their common difference is same

…(3)

We need to show that a, c, e are in GP

i.e c2 = ae

From (2), we have

c2 = bd

Putting value of

c(c + e) = e(a + c)

c2 + ce = ea + ec

c2 = ea

Thus, a, c, e are in GP.

Hence, Proved.

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