sequences , series ♡

The numbers in a sequence are called its terms.

The nth term of a sequence is denoted with $t_{n}$ . So the first term is $t_{1}$ .

sequence

A sequence can be defined in many ways

A recurrence relation is a relation that defines a sequence recursively. It means each term is defined using previous terms.

A recurrence relation must have initial conditions - these are specific values of the sequence that are defined directly.

Consider the sequence 3,5,7,9… - The recurrence relation is $t_{1} = 3$ , $t_{n} = t_{n - 1} + 2$ .

Consider the sequence 1,1,2,3,5,8,13… - The recurrence relation is $t_{1} = 1$ , $t_{2} = 1$ , $t_{n} = t_{n - 2} + t_{n - 1}$ .

An explicit formula defines every nth term directly as a function of n.

Consider the sequence 3,5,7,9… - The explicit formula is $t_{n} = 1 + 2 n$ .

Consider the sequence 2,5,10,17,26… - The explicit formula is $t_{n} = n^{2} + 1$ .

A series is a sum of some terms in a sequence.