special series

Question

The sum 1² + 2² + 3² + ... + n² equals

RankGuru · Accepted Answer

1. **Formula Recognition**: This represents the sum of the squares of the first $n$ natural numbers.
2. **Standard Formula**: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.
3. **Example Check ($n=3$)**:
   - Sum $= 1 + 4 + 9 = 14$.
   - Formula estimate: $\frac{3(4)(7)}{6} = \frac{84}{6} = 14$. Matches.

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The sum 1³ + 2³ + 3³ + ... + n³ equals

The sum 1 + 2 + 3 + ... + n equals