Understanding Sequences

A sequence is an ordered list of numbers where each number is termed as a term.
Sequences can be finite with a specific number of terms, or infinite with no defined end.
The position of a term in a sequence is its index.
A linear sequence is a sequence where the difference between each term is constant.

Arithmetic Sequences

An arithmetic sequence is a type of linear sequence where the difference between each term is the same. This is known as the common difference.
The general formula for an arithmetic sequence is a + (n-1)d, where ‘a’ is the first term, ‘n’ is the index of the term and ‘d’ is the common difference.
To find the sum of an arithmetic sequence, the formula S = 0.5n(a + l) is used. Here, ‘S’ is the sum of the sequence, ‘n’ is the number of terms, ‘a’ is the first term, and ‘l’ is the last term.

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero value known as the common ratio.
The general formula for a geometric sequence is ar^(n-1) where ‘a’ is the first term, ‘r’ is the common ratio and ‘n’ is the index of the term.
To find the sum of the first ‘n’ terms of a geometric sequence, the formula S = a(1−r^n ) / (1−r) is employed when the common ratio is not 1.

The Fibonacci sequence is a sequence where each number is the sum of the two preceding numbers, often starting with 0 and 1.

You can determine the nth term of a sequence by examining the sequence and determining the pattern or the rule being followed.
Be sure to show all workings when solving problems and always check your answers. In sequence problems, it’s advisable to work out the first few terms to avoid mistakes.