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Algebra Level 3

In the above equation, it is known that a, c, and b are 3 consecutive positive integers with a < c < b a < c < b . What is the value of a + b − c a + b - c ?


The answer is 2014.

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James R T by James R T , 21, Indonesia

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1 solution

Adrian Neacșu
Apr 17, 2014

1 + 1 n 2 + 1 ( n + 1 ) 2 = n 4 + 2 n 3 + 3 n 3 + 2 n + 1 n ( n + 1 ) = n 2 + n + 1 n ( n + 1 ) = n + 1 n − 1 n + 1 \sqrt { 1+\frac { 1 }{ { n }^{ 2 } } +\frac { 1 }{ \left( n+1 \right) ^{ 2 } } } =\frac { \sqrt { { n }^{ 4 }+{ 2n }^{ 3 }+{ 3n }^{ 3 }+2n+1 } }{ n\left( n+1 \right) } =\frac { { n }^{ 2 }+n+1 }{ n\left( n+1 \right) } =n+\frac { 1 }{ n } -\frac { 1 }{ n+1 } Then calculate the sum and take a = n , c = n + 1 a=n,c=n+1 and b = n + 2 b=n+2

10 Helpful 0 Interesting 0 Brilliant 0 Confused

oh..man once again....I missed such a nice question...I don't know why do I sometimes go wrong....anyways..a nice solution Adrian

Max B - 7 years, 1 month ago

are you sure that its n at the beginning and not 1?

1+1/n+1/(n+1). dont worry, answer is right, did it the same way.

Trevor Arashiro - 6 years, 7 months ago

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