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. ⇒2b=a+c...........(i) b, c, d are in G.P. ⇒c2=bd.........(ii) 1c,1d,1e are in A.P. ⇒2d=1c+1e.......(iii) We need to prove that a, b, c are in G.P. ⇒c2=ae Now, c2=bd=2b×d2 ⇒c2=(a+c)×cec+e [∵2d=e+cce] ⇒c2=(c+e)=ace+c2e ⇒c3+c2e=ace+c2e ⇒c3=ace ⇒c2=ae Hence proved.

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