Break them apart

Algebra Level 2

1 2 × 3 + 1 3 × 4 + 1 4 × 5 + … + 1 49 × 50 \large \frac { 1 }{ 2\times 3} +\frac { 1 }{ 3\times 4 } +\frac { 1 }{ 4\times 5 } +\ldots+\frac { 1 }{ 49\times 50 }

If the expression above can be represented in the form of a b \frac { a }{ b } ,where gcd ⁡ ( a , b ) = 1 \gcd (a,b)=1 , then find ( a + b ) 3 { (a+b) }^{ 3 } .


The answer is 50653.

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Swapnil Das by Swapnil Das , 20, India

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2 solutions

Noel Lo
May 30, 2015

1/(n*(n+1)) = 1/n - 1/(n+1). So we have 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5+ ... + 1/49 - 1/50 = 1/2 - 1/50 = 24/50 = 12/25. a=12, b=25. (a+b)^3 = 37^3 = 50653.

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Challenge student note: Nice use of telescopic sum.

Bonus question: Can you find the value of summation :

1 1.2.3 + 1 2.3.4 + 1 3.4.5 ⋯ + 1 48.49.50 = ? \dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + \dfrac{1}{3.4.5} \dots + \dfrac{1}{48.49.50} = \ ?

Nihar Mahajan - 6 years ago

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t n = 1 n ( n + 1 ) ( n + 2 ) = > 2 t n = ( n + 2 ) − n n ( n + 1 ) ( n + 2 ) = > 2 t n = 1 n ( n + 1 ) − 1 ( n + 1 ) ( n + 2 ) { t }_{ n }\quad =\quad \frac { 1 }{ n(n+1)(n+2) } \\ =>\quad 2{ t }_{ n }\quad =\quad \frac { (n+2)-n }{ n(n+1)(n+2) } \quad \\ =>\quad 2{ t }_{ n }\quad =\quad \frac { 1 }{ n(n+1) } -\frac { 1 }{ (n+1)(n+2) }

Substituting various values of n: , we get the following equations: 2 t 1 = 1 1.2 − 1 2.3 2 t 2 = 1 2.3 − 1 3.4 2{ t }_{ 1 }\quad =\quad \frac { 1 }{ 1.2 } -\frac { 1 }{ 2.3 } \\ 2{ t }_{ 2 }\quad =\quad \frac { 1 }{ 2.3 } -\frac { 1 }{ 3.4 } \quad

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2 t n = 1 n ( n + 1 ) − 1 ( n + 1 ) ( n + 2 ) 2{ t }_{ n }\quad =\quad \frac { 1 }{ n(n+1) } -\frac { 1 }{ (n+1)(n+2) }

Adding all the above equations, we get

2 ∑ t n = 1 2 − 1 ( n + 1 ) ( n + 2 ) = > ∑ t n = 1 4 − 1 2 ( n + 1 ) ( n + 2 ) 2\sum { { t }_{ n } } =\quad \frac { 1 }{ 2 } \quad -\quad \frac { 1 }{ (n+1)(n+2) } \\ =>\sum { { t }_{ n } } \quad =\quad \frac { 1 }{ 4 } \quad -\quad \frac { 1 }{ 2(n+1)(n+2) }

Substituting n = 48, we get

∑ n = 1 48 t n = 1 4 − 1 2.49.50 = 1 4 − 1 4900 = 306 1225 \sum _{ n=1 }^{ 48 }{ { t }_{ n } } \quad =\quad \frac { 1 }{ 4 } \quad -\quad \frac { 1 }{ 2.49.50 } \quad =\quad \frac { 1 }{ 4 } -\frac { 1 }{ 4900 } =\frac { 306 }{ 1225 }

This is my reply to the bonus question. I hope my method and answer are correct. @Nihar Mahajan

Manish Dash - 6 years ago

Correct analysis!

Swapnil Das - 6 years ago
Towhidd Towhidd
Jul 26, 2015

1/2x3+1/3x4+1/4x5+1/5x6+1/6x7+1/7x8+.....................+1/49x50 =1/2-1/3+1/3-1/4+1/4-1/5+.........................-1/49+1/49-1/50 =1/2-1/50 =12/25

hence, (a+b)^3=(12+25)^3=37^3=50653

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