How do I work out the nth term?

All you need is two formulas when you're looking to answer how to find the nth term. Do these come in an arithmetic and quadratic flavour? What is the nth term? Well, the number lies within a sequence of numbers, and it can be anywhere in that sequence. The right formula will save you a lot of time as a student in pinpointing this, and it can help you solve for and find n. Let's look at a common example and examine the process of answering “what is the nth term?” closely. 

Firstly, be prepared to see this type of question on the table of any maths exam you're looking to take. Below, we'll cover how to solve the nth term in the example of both an arithmetic and a geometric sequence. If you want to achieve high marks, this is crucial to know.

Understanding pattern recognition is one of the most important aspects of learning maths. It is a desirable trait to have when looking to forge a path to a lucrative career in the future. Finding the nth term manually and via formula only helps showcase and strengthen pattern recognition skills. 

An often confusing part that occurs when you look at how to find the nth term number is the word "term". The term number means the position of the number you're looking for. That means the first number is the first term, the second number is the second term, and so on. Often, students who may not be studying diligently can get confused in this area. 

Finding the nth term for an arithmetic sequence

The formula is: 

an = a1 + (n-1) d 

• an is the nth term that we’re looking for

• a1 is the first term of the sequence

• n is the term number we want to find 

• d is the common difference or jump between each step. This could be an add or subtract function. 

Finding the nth term for geometric sequences

The difference between the last sequence and this is largely that it’s a bit harder to comprehend, but again, there is a formula you can bring to the table. So, as long as you spend time on each variable, it will become clear. Geometric sequences need to be taken a bit slower because their jumps between one number to the next can be exponential and thus appear confusing at first. 

The formula this time is:

an = a1 * r (n-1)

This time, you're not looking at a consistent add or subtract but by a ratio denoted by r in this case. 

• an is the nth term we’re looking for

• a1, again, is the first term

• n is the term we want to find

• r is that common ratio

Once you start to plug in the numbers of the sequence, you will see how easy it is to find any number correctly, even the thousandth one. Remember that this is simply the beginning of finding the nth sequence. 

Breaking it down further

If you look at the formula itself, you can break it down when you have enough sequence numbers to build out the formula for the sequence itself. This is another way to showcase your maths prowess in a new format. It will also help you start on your journey of resolving even more complex sequences that you will need to learn further down the line.