Child Labor #8

Algebra Level 3

If a , b , c , d , e a, b, c, d,e form a harmonic progression, then a b + b c + c d + d e ab + bc + cd + de is independent of:

b , c and d b,c \text{ and } d d only d \text{ only} b only b \text{ only} c only c \text{ only}

Rajen Kapur by Rajen Kapur , 72, India

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1 solution

Rajen Kapur
Oct 7, 2015

a b + b c + c d + d e ab + bc + cd + de = a b c ( 1 c + 1 a ) + c d e ( 1 e + 1 c ) = abc(\dfrac{1}{c} + \dfrac{1}{a}) + cde(\dfrac{1}{e} + \dfrac{1}{c}) = a b c ( 2 b ) + c d e ( 2 d ) = abc(\dfrac{2}{b}) + cde(\dfrac{2}{d}) = 2 a c + 2 c e = 2 a c e ( 1 e + 1 a ) = 2 a c e ( 2 c ) = 4 a e =2ac + 2ce=2ace(\dfrac{1}{e} + \dfrac{1}{a})=2ace(\dfrac{2}{c})=4ae . Hence the conclusion that the expression is independent of b, c, and d.

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