Q.If the middle number of the first \((2n + 1)\) consecutive natural numbers is \(\frac{n + 103}{3}\), then find the value of \(n\).

For any value of \(n\), \((2n + 1)\) is an odd number. \(\therefore\) The middle term of the first \((2n + 1)\) consecutive natural numbers is the \(\frac{(2n + 1) + 1}{2}\)th term = \(\frac{(2n + 2)}{2}\)th term = \((n + 1)\)th term \(\therefore\) The middle number is \((n + 1)\) According to the question, \(n + 1 = \frac{n + 103}{3}\) i.e., \(3n + 3 = n + 103\) i.e., \(3n - n = 103 - 3\) i.e., \(2n = 100\) i.e., \(n = 50\) \(\therefore\) The required value of \(n\) is \(50\)
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