100 followers problem!

Number Theory Level 2

Find the value of N \text{N} .

N \text{N} = 1 3 + 2 3 + 3 3 + 4 3 … + 9 9 3 + 10 0 3 1^3 + 2^3 + 3^3 + 4^3 \ldots +99^3 + 100^3 .


The answer is 25502500.

Arpit MIshra by Arpit MIshra , 21, India

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3 solutions

Rama Devi
May 14, 2015

We know that the sum of cubes of first n natural numbers is {n(n+1)/2}^2. By substituting the value of n as 100, we get the answer as 25502500.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Arpit MIshra
Apr 24, 2015

N = 1 3 + 2 3 + 3 3 + 4 3 … + 10 0 3 \text{N}=1^3+2^3+3^3+4^3 \ldots + 100^3 .

= ( 1 + 2 + 3 … + 100 ) 2 (1+2+3 \ldots +100)^2 .

= ( ( n ) ( n + 1 ) 2 ) 2 (\dfrac{(n)(n+1)}{2})^2 .

= ( ( 100 ) ( 100 + 1 ) 2 ) 2 (\dfrac{(100)(100+1)}{2})^2 .

= ( 100 ( 101 ) 2 ) 2 (\dfrac{100(101)}{2})^2 .

= ( 50 × 101 ) 2 (50 \times 101)^2 .

= ( 5050 ) 2 (5050)^2 .

= 25502500 \boxed{25502500} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

The sum of the first n n cubes is the square of the sum of the first n n positive integers. Therefore the answer is

[ 100 2 ( 101 ) ] 2 = \left[\dfrac{100}{2}(101)\right]^2= 25502500 \color{#D61F06}\boxed{25502500}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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