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Question

If a, b, c are in A.P.; b, c, d are in G.P. and 1c,1d,1e are in A.P. prove that a, c, e are in G.P.

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Solution

a, b, c are in A.P.

ba=cb2b=a+c

b=a+c2(1)

b, c, d are in G.P.

cb=dcc2=bd(2)

Also, 1c, 1d, 1e are in A.P.

1d1c=1e1d2d=1c+1e

2d=c+eced=2cec+e(3)

Putting value of equation (1) and (3) in (2)

c2=(c+a2)(2cec+e)=ce[c+a]c+e

c2[c+e]=ec[c+a]

c2+ce=ce+ae

c2=ae

which shows that a, c, e are in G.P.


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