Summation Formulas: Common Sums and Examples

Summation Formulas

Summation formulas make it easier to add a sequence of numbers without writing every term separately. The symbol Σ (sigma) is used to represent a sum, and it is especially useful when working with sequences, series, and algebraic expressions.

What Is Summation Notation?

Summation notation provides a short way to write a series of additions. A common form is:

∑_i=1ⁿ a_i = a₁ + a₂ + a₃ + ... + a_n

Here, i is the index of summation, 1 is the starting value, n is the ending value, and a_i is the expression being added.

Sum of the First n Natural Numbers

The sum of the first n positive integers is:

∑_i=1ⁿ i = n(n + 1) / 2

For example, the sum from 1 to 10 is:

10(10 + 1) / 2 = 55

Sum of Squares

When each integer is squared before adding, use:

∑_i=1ⁿ i² = n(n + 1)(2n + 1) / 6

For example:

1² + 2² + 3² + 4² = 30

Sum of Cubes

The sum of the cubes of the first n positive integers is:

∑_i=1ⁿ i³ = [n(n + 1) / 2]²

This can also be written as:

1³ + 2³ + ... + n³ = [n(n + 1) / 2]²

Sum of a Constant

If the same constant is added n times, the summation is simply:

∑_i=1ⁿ c = nc

For example, if c = 7 and there are 5 terms:

7 + 7 + 7 + 7 + 7 = 5(7) = 35

Arithmetic Series Sum Formula

An arithmetic series has a constant difference between consecutive terms. If a is the first term, d is the common difference, and n is the number of terms, its sum is:

S_n = n/2 [2a + (n − 1)d]

If the first term and last term are known, the simpler form is:

S_n = n(a + l) / 2

where l is the last term.

Geometric Series Sum Formula

A geometric series has a constant ratio between consecutive terms. For a first term a, common ratio r, and n terms:

S_n = a(1 − rⁿ) / (1 − r)

This formula applies when r ≠ 1. If r = 1, every term is equal to a, so:

S_n = na

Infinite Geometric Series Formula

An infinite geometric series has a finite sum only when the absolute value of its common ratio is less than 1:

|r| < 1

When this condition is satisfied, the sum is:

S_∞ = a / (1 − r)

If |r| ≥ 1, the infinite geometric series does not have a finite sum.

Useful Properties of Summation

Summation can be separated across addition:

∑(a_i + b_i) = ∑a_i + ∑b_i

A constant factor can also be taken outside the summation:

∑(c a_i) = c∑a_i

These properties are useful for simplifying longer algebraic sums before calculating them.

Common Summation Formulas

Type	Formula
First n natural numbers	n(n + 1) / 2
Sum of squares	n(n + 1)(2n + 1) / 6
Sum of cubes	[n(n + 1) / 2]²
Constant	nc
Arithmetic series	n[2a + (n − 1)d] / 2
Arithmetic series using first and last terms	n(a + l) / 2
Finite geometric series	a(1 − rⁿ) / (1 − r)
Infinite geometric series	a / (1 − r), \|r\| < 1

Summation Formulas: Common Sums and Examples