JEE Main 2016 (27)

Algebra Level 3

For some natural number n n , the sum of first n n natural numbers is 240 less than the sum of the first ( n + 5 ) (n+5) natural numbers. Then n n itself is the sum of how many natural numbers starting with 1?

Try my set JEE Main 2016 .
10 6 None of these choices 9 7 8

Shivam Jadhav by Shivam Jadhav , 22, India

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1 solution

Dev Sharma
Mar 21, 2016

The sum of first k natural number is given by k ( k + 1 ) 2 \frac{k(k+1)}{2} .

According to question,

n ( n + 1 ) 2 + 240 = ( n + 5 ) ( n + 6 ) 2 \frac{n(n+1)}{2} + 240 = \frac{(n+5)(n+6)}{2}

n 2 + n + 480 = n 2 + 11 n + 30 n^2 + n + 480 = n^2 + 11n + 30

n = 45 n = 45

Let n n be the sum of first x x natural numbers, then

x ( x + 1 ) 2 = 45 \frac{x(x+1)}{2} = 45

x = 9 x = 9

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