1² + 3² + 5² + .................. + 31² = ?

প্রশ্ন: 1² + 3² + 5² + .................. + 31² = ?

সমাধান:
আমরা জানি,
1² +2²+ 3²+ ......... + n²= {n(n +1)(2n + 1)}/6
1² +2²+ 3²+ ......... + 31² = {31(31 +1)(2×31 + 1)}/6
= (31 × 32× 63)/6
= 10416

আবার
2² + 4² + 6² + ............. + 30² = 2²(1² + 2² + ³² + .............. + 15²)
= 4 {15(15 +1)(2 × 15 + 1)}/6
= 4960

এখন
1² + 3² + 5² + .................. + 31² = (1² + 2²+ 3²+ ......... + 31²) - (2² + 4² + 6² + ............. + 30²)
= 10416 - 4960
= 5456