Summation Formulas

Standard formulas for evaluating finite sums of polynomial sequences. These formulas are essential for calculating sums efficiently without computing each term individually.

Sigma Notation

Summation notation provides a compact way to represent the sum of a sequence:

$$\sum_{r=1}^{n} a_r = a_1 + a_2 + a_3 + \ldots + a_n$$

Where:

  • $r$ is the index of summation
  • $1$ is the lower limit
  • $n$ is the upper limit
  • $a_r$ is the general term

Standard Formulas

Sum of First n Natural Numbers (Arithmetic Series)

$$\sum_{r=1}^{n} r = 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$$

Proof by induction or by pairing terms (Gauss's method): $1 + n = 2 + (n-1) = 3 + (n-2) = \ldots = n+1$, with $n/2$ such pairs.

Sum of Squares

$$\sum_{r=1}^{n} r^2 = 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}$$

Sum of Cubes

$$\sum_{r=1}^{n} r^3 = 1^3 + 2^3 + 3^3 + \ldots + n^3 = \frac{n^2(n+1)^2}{4}$$

[!important] Special Property The sum of cubes equals the square of the sum of the first $n$ natural numbers: $$\sum_{r=1}^{n} r^3 = \left(\sum_{r=1}^{n} r\right)^2 = \left[\frac{n(n+1)}{2}\right]^2$$

Properties of Summation

Linearity

$$\sum_{r=1}^{n} (ca_r + db_r) = c\sum_{r=1}^{n} a_r + d\sum_{r=1}^{n} b_r$$

Change of Index

$$\sum_{r=m}^{n} a_r = \sum_{r=1}^{n} a_r - \sum_{r=1}^{m-1} a_r$$

Worked Examples

Example 1: Basic Application

Find $\sum_{r=1}^{10} r$: $$\sum_{r=1}^{10} r = \frac{10(11)}{2} = 55$$

Example 2: Expanded Polynomial

Find $\sum_{r=1}^{10} (r+7)(r-1)$:

First expand: $(r+7)(r-1) = r^2 + 6r - 7$

Then apply linearity: $$\sum_{r=1}^{10} (r^2 + 6r - 7) = \sum r^2 + 6\sum r - \sum 7$$ $$= \frac{10(11)(21)}{6} + 6 \cdot \frac{10(11)}{2} - 7(10)$$ $$= 385 + 330 - 70 = 645$$

Example 3: Sum of Odd Squares

Find $1^2 + 3^2 + 5^2 + \ldots$ to 20 terms.

The $r$-th odd number is $2r-1$, so we need: $$\sum_{r=1}^{20} (2r-1)^2 = \sum_{r=1}^{20} (4r^2 - 4r + 1)$$ $$= 4\sum r^2 - 4\sum r + \sum 1$$ $$= 4 \cdot \frac{20(21)(41)}{6} - 4 \cdot \frac{20(21)}{2} + 20$$

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