a + b

Algebra Level 3

1 2007 ⋅ 2008 + 1 2008 ⋅ 2009 + 1 2009 ⋅ 2010 + 1 2010 = a b \frac{1}{2007\cdot2008} + \frac{1}{2008\cdot2009} + \frac{1}{2009\cdot2010} + \frac{1}{2010} = \frac{a}{b}

Find value of a+b

2011 2009 2008 2007 2010

Yosua Sibuea by Yosua Sibuea , 22, Indonesia

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2 solutions

Rawan Medhat
Nov 13, 2015

By taking 1 2010 \frac { 1 }{ 2010 } common factor :

1 2010 ( 1 + 1 2009 ) + 1 2008 ⋅ 2009 + 1 2008 ⋅ 2007 \frac { 1 }{ 2010 } \left( 1+\frac { 1 }{ 2009 } \right) +\frac { 1 }{ 2008\cdot 2009 } +\frac { 1 }{ 2008\cdot 2007 }

1 2010 ( 1 + 2009 2009 ) + 1 2008 ⋅ 2009 + 1 2008 ⋅ 2007 \frac { 1 }{ 2010 } \left( \frac { 1+2009 }{ 2009 } \right) +\frac { 1 }{ 2008\cdot 2009 } +\frac { 1 }{ 2008\cdot 2007 }

1 2010 ( 2010 2009 ) + 1 2008 ⋅ 2009 + 1 2008 ⋅ 2007 \frac { 1 }{ 2010 } (\frac { 2010 }{ 2009 }) +\frac { 1 }{ 2008\cdot 2009 } +\frac { 1 }{ 2008\cdot 2007 }

1 2009 + 1 2008 ⋅ 2009 + 1 2008 ⋅ 2007 \frac { 1 }{ 2009 } +\frac { 1 }{ 2008\cdot 2009 } +\frac { 1 }{ 2008\cdot 2007 }

By repeating these steps

a b = 1 2007 \frac { a }{ b } =\frac { 1 }{ 2007 }

a+b = 2008

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