Sequences

Sequence problems typically include a list of numbers, or they’ll describe a process to calculate a list of numbers. Then they’ll ask you something like the sum of the numbers or the value of the $n$th term in the list.

Pattern sequences

Here’s an example of a very common sequence question type:

List A contains all numbers that are multiples of 5 and greater than 10, in ascending order.

What is the 4th number of the list?

Start by writing out the list, and this version becomes pretty straightforward: 15, 20, 25, 30, 35… so the answer is 30. Pay close attention to any conditions like greater than and ascending so you don’t make any careless mistakes from reading too quickly!

Here’s another common version:

Sequence A repeats with the following pattern: 3, 6, 2, 1, 3, 6, 2, 1, 3, 6…

What is the 40th number in the sequence?

The list’s pattern repeats the four numbers: 3, 6, 2, 1. Every fourth number in this list is 1, for example, the 4th, 8th, 12th, 16th, and so on. The question asks us for the 40th number in the sequence, and since 40 is a multiple of 4, the 40th number must also be 1.

Sometimes you’ll see a variation on this question that asks for the sum of all the terms from the 40th to the 43rd term. Since we know the 40th number is 1, the 41st will be 3, 42nd will be 6, and the 43rd will be 2. The sum is $3+6+2+1=12$.

There’s a shortcut we could take for this question. Since the pattern consists of four repeating numbers, any four terms in a row will be the sum of those numbers. The answer will be the same, regardless of it it’s asking for the 40th to 43rd, 41st to 44th, 42nd to 45th, or any other four adjacent terms.

Calculated sequences

This kind of sequence problem is a bit more complicated. These problems usually give you one term in the sequence, along with a method to calculate the other terms.

For example:

The 5th term in Sequence A is -12. Every previous number is 3 greater than the number following it. What is the 1st term in this sequence?

A good strategy is to draw out a table with term positions and values. Start with the information provided, and complete the table step by step until you get to the term you need.

If the 4th term is 3 greater than the 5th term, then the 4th term must be -9. The 3rd must be -6. And so on:

The first term in the sequence is 0.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: