Sigma Notation Math Example 4

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

Example 4

medium
Rewrite โˆ‘j=1n(3j2+2jโˆ’1)\displaystyle\sum_{j=1}^{n}(3j^2 + 2j - 1) using linearity of summation.

Solution

  1. 1
    Split: 3โˆ‘j=1nj2+2โˆ‘j=1njโˆ’โˆ‘j=1n13\sum_{j=1}^n j^2 + 2\sum_{j=1}^n j - \sum_{j=1}^n 1.
  2. 2
    Apply formulas: 3โ‹…n(n+1)(2n+1)6+2โ‹…n(n+1)2โˆ’n3\cdot\frac{n(n+1)(2n+1)}{6} + 2\cdot\frac{n(n+1)}{2} - n.
  3. 3
    Simplify: n(n+1)(2n+1)2+n(n+1)โˆ’n\frac{n(n+1)(2n+1)}{2} + n(n+1) - n.

Answer

n(n+1)(2n+1)2+n(n+1)โˆ’n\frac{n(n+1)(2n+1)}{2} + n(n+1) - n
Linearity of sigma: โˆ‘(af+bg+c)=aโˆ‘f+bโˆ‘g+cโ‹…n\sum(af+bg+c) = a\sum f + b\sum g + c\cdot n. Pull constants outside and use the standard formulas for โˆ‘k\sum k and โˆ‘k2\sum k^2.

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 2 medium

Write [formula] in sigma notation and evaluate the closed form for [formula].

Example 3 easy

Evaluate [formula].

โ† All Sigma Notation Examples Sigma Notation Concept