The numbers 1, 2, 3, 4, 5, 6 are called the natural numbers.

By adding the terms to form a series, 1 + 2 + 3 + 4 + 5 + 6 + ... n we have the sum of the natural numbers, which is written as:

Since this is an AP where the first term a = 1 and the common difference d = 1, we can evaluate the sum to be:

If we square the natural numbers and add the terms to obtain the series

1² + 2² + 3² + 4² + 5² + 6² + ... n²

For the sum of the square numbers, we have:

Finally, the sum of the cubes of the first n natural numbers is:

Recurrence relations

A recurrence relation is when there is a link between two or more of the terms of a sequence. The first few terms are usually given.

For example:

if we the first two terms of a sequence are T₁ = 2 and T₂ = 6 and we know that T_n+2 = T_n+1 + T_n, then we can find the next two terms as follows.

When n = 1, T₃ = T₂ + T₁ = 6 + 2 = 8

When n = 2, T₄ = T₃+ T₂ = 8 + 6 = 14

This gives us the sequence

2, 6, 8, 14, 22, 36,...

which is a variation on the Fibonacci sequence.

Periodicity

A sequence that repeats its terms in the same order after a certain number of terms is called periodic or cyclic. In trigonometry, we see how the sine and cosine graphs show a range between 1 and -1, and they repeat every 2π radians.

Therefore, the sequence T = cos(nπ/2) has the terms 0, -1, 0, 1,... which repeat every four terms.

Oscillation

A sequence such as T = (-1)ⁿ has the terms -1, 1, -1, 1,...

This oscillates finitely between -1 and 1.

However the sequence T = (-10)ⁿ has the terms -10, 100, -1000, 10 000,...

This oscillates infinitely between −∞ and ∞.