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JEE Main202227 Jun 2022Morning ShiftMathematicsSequences and SeriesActual

If the sum of the first ten terms of the series 1 5 + 2 65 + 3 325 + 4 1025 + 5 2501 + … . is m n , where m and n are co-prime numbers, then m + n is equal to ______.

Correct answer

0

Step-by-step solution

For the series 1 5 + 2 65 + 3 325 + 4 1025 + 5 2501 + … … T n = n 4 n 4 + 1 = n 2 n 2 + 1 2 - ( 2 n ) 2 = n 2 n 2 + 2 n + 1 2 n 2 - 2 n + 1 = 1 4 1 2 n 2 - 2 n + 1 - 1 2 n 2 + 2 n + 1 Now S 10 = ∑ n = 1 10   T n = ∑ n = 1 10 1 4 1 2 n 2 - 2 n + 1 - 1 2 n 2 + 2 n + 1 = 1 4 1 1 - 1 5 + 1 5 - 1 13 + 1 13 … … … + 1 181 - 1 221 = 1 4 1 - 1 221 = 55 221 = m n Hence, m + n = 55 + 221 = 276

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