Overview

Power sums are frequently embedded in counting and algebraic simplifications. Memorize the core formulas and recognize when to subtract ranges.

Key Ideas

• $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$.
• $1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$.
• $1^3 + 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$.

Core Skills

Subtract Ranges

Compute $a+\cdots+b$ as $1+\cdots+b$ minus $1+\cdots+(a-1)$.

Spot Cubic Sum Squares

Remember $1^3+\cdots+n^3$ is the square of the triangular sum. This shortcut appears often.

Factor Before Plugging In

Simplify expressions like $n(n+1)$ early to reduce arithmetic errors.

Worked Example

Compute $51 + 52 + \cdots + 100$.

Use subtraction of sums: $1 + 2 + \cdots + 100 = \frac{100 \cdot 101}{2} = 5050$ and $1 + 2 + \cdots + 50 = \frac{50 \cdot 51}{2} = 1275$. So the answer is $5050 - 1275 = 3775$.

More Examples

Example 1: Squares

Compute $1^2+2^2+\cdots+20^2$.

$\frac{20\cdot 21\cdot 41}{6} = 2870$.

Example 2: Cubes

Compute $1^3+2^3+\cdots+10^3$.

$\left(\frac{10\cdot 11}{2}\right)^2 = 55^2 = 3025$.

Example 3: Range of Squares

Compute $8^2+9^2+\cdots+15^2$.

Use subtraction: $(1^2+\cdots+15^2) - (1^2+\cdots+7^2)$.

Strategy Checklist

• Decide if the problem is a full sum or a range.
• Use subtraction for partial ranges.
• Remember the cube sum shortcut.
• Keep arithmetic exact until the last step.

Common Pitfalls

• Forgetting that $1^3 + \cdots + n^3$ equals the square of the triangular sum.
• Using the even/odd sum formulas with the wrong endpoint.
• Plugging the wrong value of $n$ after a range subtraction.

Practice Problems