Find the sum to n terms: 3×1² + 5×2² + 7×3² + …

The given series is 3×1² + 5×2² + 7×3² + … nᵗʰ term, aₙ = (2n + 1)n² = 2n³ + n²

∴ Sₙ = Σaₙ = Σ(2k³ + k²) = 2Σk³ + Σk²

= 2[(n(n + 1)/2)²] + [n(n + 1)(2n + 1)/6]

= n²(n + 1)² / 2 + n(n + 1)(2n + 1) / 6

= (n(n + 1)/2)[n(n + 1)/2 + (2n + 1)/3]

= (n(n + 1)/2)[(3n² + 3n + 2n + 1)/3]

= (n(n + 1)/2)[(3n² + 5n + 1)/3]

= n(n + 1)(3n² + 5n + 1) / 6

∴ Sₙ = n(n + 1)(3n² + 5n + 1) / 6