The sum, Σₙ₌₁⁷ n( n + 1 )( 2n + 1 )/4 is equal to ________.

Solution

Maths sir · 4 steps

Open up the product into powers of n, then use the three standard sums up to 7.

1. Expand the term. Multiply out the product: n(n+1)(2n+1) = 2n³ + 3n² + n. Each power of n can now be summed separately.

   Σₙ₌₁⁷ n(n+1)(2n+1)/4 = 1/4(2Σ n³ + 3Σ n² + Σ n)

2. Standard sums. For n = 7: Σ n = (7 × 8)/2 = 28, Σ n² = (7 × 8 × 15)/6 = 140, and Σ n³ = 28² = 784.
3. Put in numbers. 2 × 784 + 3 × 140 + 28 = 1568 + 420 + 28 = 2016. Dividing by 4 gives 504.
4. So. The sum is 504.

The common slip: 2016. Forgetting the factor 1/4 outside the bracket gives 2016, not 504.

The idea to keep. When the general term is a polynomial in n, expand it fully and use Σ n, Σ n², Σ n³. Remember Σ n³ is just (Σ n)².

Answer from the NTA final answer key. Solution by PrepWiser.