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Black Advanced Problems in Mathematics
Exam
Black Advanced Problems in Mathematics
Subject
Mathematics
Chapter
Sequences and Series

Black Advanced Problems in Mathematics · Mathematics · Sequences and Series

Let a, b, c, d, e are non-zero and distinct positive real numbers....

Let a,b,c,d,ea, b, c, d, e are non-zero and distinct positive real numbers. If a,b,ca, b, c are in A.P.; b,c,db, c, d are in G.P. and c,d,ec, d, e are in H.P., then a,c,ea, c, e are in:

Options

  1. AA.P.
  2. BG.P.
  3. CH.P.
  4. DNothing can be said

Correct answer

C. H.P.

Step-by-step solution

Given a,b,ca, b, c are in A.P.   ⟹  b=a+c2c,d,e\implies b = \frac{a+c}{2}c, d, e are in H.P.   ⟹  d=2cec+eb,c,d\implies d = \frac{2ce}{c+e}b, c, d are in G.P.   ⟹  c2=bd\implies c^2 = bd Substituting bb and dd in the G.P. condition: c2=(a+c2)(2cec+e)c2=ce(a+c)c+ec(c+e)=e(a+c)c2+ce=ae+cec2=aec^2 = \left(\frac{a+c}{2}\right)\left(\frac{2ce}{c+e}\right)c^2 = \frac{ce(a+c)}{c+e}c(c+e) = e(a+c)c^2 + ce = ae + cec^2 = ae This means a,c,ea, c, e are in G.P.
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