Sigma Notation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Write 12+22+32+โ‹ฏ+n21^2 + 2^2 + 3^2 + \cdots + n^2 in sigma notation and evaluate the closed form for n=10n = 10.

Solution

  1. 1
    In sigma notation: โˆ‘k=1nk2\displaystyle\sum_{k=1}^{n} k^2.
  2. 2
    Closed-form formula: โˆ‘k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}.
  3. 3
    For n=10n = 10: 10โ‹…11โ‹…216=23106=385\frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385.

Answer

โˆ‘k=1nk2=n(n+1)(2n+1)6\displaystyle\sum_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6}; for n=10n=10: 385385
Sigma notation concisely represents this sum. The closed-form formula is a standard result that avoids adding all terms individually โ€” essential for large nn and for Riemann sum computations.

About Sigma Notation

Sigma notation uses the Greek letter ฮฃ to express the sum of many terms compactly. The expression โˆ‘i=1nai\sum_{i=1}^{n} a_i means 'add up aia_i for every integer ii from 1 to nn.' For example, โˆ‘i=14i2=1+4+9+16=30\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30.

Learn more about Sigma Notation โ†’

More Sigma Notation Examples

Example 1 easy

Expand and evaluate [formula].

Example 3 easy

Evaluate [formula].

Example 4 medium

Rewrite [formula] using linearity of summation.

โ† All Sigma Notation Examples Sigma Notation Concept