Sequences

Sequence problems usually give you a list of numbers (or a rule for generating a list). Then they ask for something like the sum of the numbers or the value of the $n$th term.

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:

15, 20, 25, 30, 35, …

So the 4th number is 30. Pay close attention to conditions like greater than and ascending so you include the right numbers in the right order.

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 pattern repeats every four terms: 3, 6, 2, 1.

That means every 4th term is 1 (the 4th, 8th, 12th, 16th, and so on). Since 40 is a multiple of 4, the 40th term is also 1.

Sometimes you’ll see a variation that asks for the sum of several consecutive terms, such as the 40th through the 43rd terms. Since the 40th term is 1, the next terms follow the pattern:

• 40th: 1
• 41st: 3
• 42nd: 6
• 43rd: 2

So the sum is $1+3+6+2=12$.

There’s also a shortcut here: because the pattern has four repeating numbers, any four consecutive terms will have the same sum. So you’d get the same total whether the question asks for the 40th-43rd, 41st-44th, 42nd-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 rule for finding 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 a table with term positions and values. Start with the information you’re given, then fill in the missing terms step by step until you reach the term you need.

“Every previous number is 3 greater than the number following it” means:

• 4th term = 5th term + 3
• 3rd term = 4th term + 3
• and so on

So if the 5th term is -12, then the 4th term is -9. The 3rd term is -6, and you keep adding 3 as you move backward:

The 1st 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: