Enough manipulations?

Algebra Level 3

If a , b , c a,b,c and d d follows a harmonic progression (in that order), which of these answer choices equals to a b + b c + c d ab + bc + cd ?

3 a c 3ac ( a + b ) ( c + d ) (a+b)(c+d) None of these choices 3 a d 3ad

Palepu Tarun Sathwik by Palepu Tarun Sathwik , 20, India

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2 solutions

Arjen Vreugdenhil
Sep 18, 2015

A more elementary solution: a = 1 n ; b = 1 n + m ; c = 1 n + 2 m ; d = 1 n + 3 m . a = \frac{1}{n};\;b=\frac{1}{n+m};\;c=\frac{1}{n+2m};\;d=\frac{1}{n+3m}. The given sum is 1 n ( n + m ) + 1 ( n + m ) ( n + 2 m ) + 1 ( n + 2 m ) ( n + 3 m ) \frac{1}{n(n+m)}+\frac{1}{(n+m)(n+2m)}+\frac{1}{(n+2m)(n+3m)} = ( n + 2 m ) ( n + 3 m ) + n ( n + 3 m ) + n ( n + m ) n ( n + m ) ( n + 2 m ) ( n + 3 m ) =\frac{(n+2m)(n+3m)+n(n+3m)+n(n+m)}{n(n+m)(n+2m)(n+3m)} = 3 n 2 + 9 n m + 6 m 2 n ( n + m ) ( n + 2 m ) ( n + 3 m ) = 3 ( n + m ) ( n + 2 m ) n ( n + m ) ( n + 2 m ) ( n + 3 m ) =\frac{3n^2+9nm+6m^2}{n(n+m)(n+2m)(n+3m)}=\frac{3(n+m)(n+2m)}{n(n+m)(n+2m)(n+3m)} = 3 n ( n + 3 m ) = 3 a d . =\frac{3}{n(n+3m)}=3ad.

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Abhinav Whizkid
Aug 28, 2015

we know that if A1,A2,A3......An are in HP then : A1A2+A2A3+A3A4+.......+An-1An=(n-1)A1An

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