A number theory problem by Chetana Sudhir

Number Theory Level 3

For some natural no. 'n', the sum of the 1st n natural nos. is 240 less than the sum of the first (n+5) natural nos. Then n itself is the sum of how many natural nos. starting with 1? Courtesy: AMTI Screening Test


The answer is 9.

Chetana Sudhir by Chetana Sudhir , 21, India

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3 solutions

Harshita Moondra
Dec 5, 2014

since sum of first n natural nos. is 240 less than sum of the first (n+5) natural nos. So, n+5+n+4+n+3+n+2+n+1=240 this implies, n=45 now using Gauss's formula, x(x+1)=45 this implies x=9 or x=-10 x can't be -10 so x=9.

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William Isoroku
Aug 26, 2014

Use Gauss's formula for the sum : n(n+1)/2=[(n+5)(n+5+1)/2] -240. Then n is 45. Use Gauss' formula again (let's use variable A instead of n): A(A+1)/2=45. And A is 9.

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Chetana Sudhir
Aug 23, 2014

The sum of the first n natural nos. is given by the formula n(n+1)/2. Therefore the equation is: (n(n+1)/2)+240 = (n+5)(n+6) By simplifying, we get, ((n^2 + n)+480)/2 = n^2 + 11n + 30. Finally, we get n = 45. n (n+1)/2 = 45 if it is also the sum of consecutive natural nos. starting with 1. (PS: Dont confuse this n with the previous n which is 45 itself.). Here n = 9. Hence 45 is the sum of the first 9 natural nos.

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First equation should be n ( n + 1 ) 2 + 240 = ( n + 5 ) ( n + 6 ) 2 \frac{n(n+1)}{2}+240=\frac{(n+5)(n+6)}{2}

Edward Jiang - 6 years, 9 months ago

Good solution bro and whoever you share my name!!!!!!!!!

Sudhir Aripirala - 6 years, 4 months ago

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