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Sequences And Series

Question
CBSEENMA11015640

If straight x space equals space sum from straight n equals space 0 to infinity of straight a to the power of straight n comma space straight y space equals space sum from straight n equals 0 to infinity of straight b to the power of straight n comma space sum from straight n equals 0 to infinity of straight c to the power of straight n where a, b, c are in A.P. and |a| < 1, |b|<1, |c|< 1, then x, y, z are in

  • G.P.

  • A.P.

  • Arithmetic − Geometric Progression

  • H.P.

Solution

D.

H.P.

straight x space equals sum from straight n equals 0 to infinity of space straight a to the power of straight n space equals space fraction numerator 1 over denominator 1 minus straight a end fraction space space space space space space space space space straight a equals space 1 minus 1 over straight x
straight y space equals sum from straight n equals 0 to infinity of space straight b to the power of straight n space space space equals fraction numerator 1 over denominator 1 minus straight b end fraction space space space space space space space space straight b equals space space 1 minus 1 over straight y
straight z space equals space sum from straight n equals space 0 space to infinity of straight c to the power of straight n space equals space fraction numerator 1 over denominator 1 minus straight c end fraction space space space space space space space space space straight c space equals space 1 minus 1 over straight z
straight a comma straight b comma space straight c space are space in space straight A. straight P.
2 straight b space equals space straight a plus straight c
2 open parentheses 1 minus 1 over straight y close parentheses space equals 1 minus 1 over straight x plus 1 minus 1 over straight y
2 over straight y space equals space 1 over straight x plus 1 over straight z
rightwards double arrow space straight x comma straight y comma straight z space are space in space straight H. straight P.