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For any real number x, let [x] be the largest integer less than or equal to x. If \(\sum_{n=1}^N \left[ \frac{1}{5} + \frac{n}{25} \right] = 25\) then N is

Updated On: Jan 15, 2026
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Solution and Explanation

The given summation is: 

\[\sum_{n=1}^N \left[ \frac{1}{5} + \frac{n}{25} \right] = 25\]

The term inside the brackets can be rewritten as: \(\frac{1}{5} + \frac{n}{25} = \frac{5n + 1}{25}\). We are interested in the floor of this expression, denoted by \(\left[ \frac{5n + 1}{25} \right]\). 

For values of \(n\) from 1 to 19, the floor is 0. 

For values of \(n\) from 20 to 44, the floor is 1. 

We need to find the value of \(N\) for which the sum of these floor values equals 25. 

The sum can be broken down as follows: \(\sum_{n=1}^N \left[ \frac{5n + 1}{25} \right] = \sum_{n=1}^{19} 0 + \sum_{n=20}^{44} 1 = 0 + (44 - 20 + 1) \times 1 = 0 + 25 = 25\)

Therefore, the value of \(N\) that satisfies the equation \(\sum_{n=1}^N \left[ \frac{1}{5} + \frac{n}{25} \right] = 25\) is \(N = 44\).

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