Definition and finding the nth term

Understanding the Concept of nth term

A sequence refers to an ordered list of numbers. Each number in the sequence is referred to as a term.
The position of a term in a sequence is usually given a variable like n. The first term is where n = 1, the second term where n = 2, and so on.
The nth term of a sequence is a formula that enables us to find any term of the sequence without having to go up from one term to the next.
The nth term rule is a pattern, embedded in a formula, that allows for the calculation of any corresponding term in the sequence with the known value of n.

Steps to Find the nth term

Step 1: Investigate the sequence for a pattern. Look at the differences between successive terms.
Step 2: If the difference between the terms is constant, the sequence is linear. A linear sequence will have an nth term of the form an + b.
Step 3: The coefficient of n, or a, is the common difference in the sequence.
Step 4: To find the constant/bias term, substitute any term number and its corresponding term value into the equation an + b.

Example

Consider the sequence: 3, 6, 9, 12, 15
Step 1: The difference between the terms is constant and equal to +3.
Step 2: This is a linear sequence, so the nth term will be of the form an + b.
Step 3: The coefficient of n is the same as the common difference. Therefore, a = +3 and the nth term becomes 3n + b.
Step 4: Substitute a known value of n and its term into 3n + b. Using n = 1 and term value 3, the equation becomes 3*1 + b = 3. Solving this gives b = 0. Hence, the nth term is 3n.

Essential Tips

The concept of nth term involves understanding patterns and rules in sequenced numbers.
Practising with different sequences enhances proficiency in finding nth terms.
If a sequence does not have a linear pattern, methods of finding the nth term may be more complex, such as quadratic or geometric sequences.

In Summary

Finding the nth term involves identifying patterns in sequences and creating linear equations.
The nth term rule makes it possible to calculate any term in the sequence without iterating through every preceding term.