A number sequence

4KM and 4KJ have been looking at number sequences...

Number Sequences and Patterns

One student replied to a comment I wrote. The class had looked at the Fibonacci Sequence I had shared and were able to work out the next number in...

0, 1, 1, 2, 3, 5, 8, 13, 21, …

They worked out, for example,  8 came from adding the previous two numbers, i.e. 3 + 5. 13 came from adding 5 and 8. 21 came from 8 and 13. They worked out the next number would be...

13 + 21 = 34

I shared a much harder sequence of numbers I found. I think many adults might have a problem solving this one...

15, 29, 56, 108, 208, ___

Given a choice of four possible next numbers, which do you think comes next...

a) 386   (b) 400   (c)  416   (d) 438

I gave the answer as (b) 400

Why is this so?

Looking at the numbers, I first noticed each number was roughly double the previous...

15, 30, 60, 120, 240

I then looked at the difference between the doubling and the sequence number...

15, 30-1, 60-4, 120-12, 240-32

But how could we work out the pattern? The number we subtract changes. Here is the pattern...

15, 2 x 15 - 1 x 1, 4 x 15 - 2 x 2, 8 x 15 - 3 x 4, 16 x 15 - 4 x 8


1)  the number to multiply the 15 doubles each time

2) the first number in the subtracted multiplication goes up one each time

3)  the second number in the subtracted multiplication doubles each time

Using these patterns, the next number in the series would use the equation...

32 x 15 - 5 x 16 = 480 - 80 = 400

If a student understands how I worked out the sequence, what number comes after 400? Leave your answer in the comments.

Is this a case of being as clear as mud? 🙂

Schools and students have permission to use this graphic for non-commercial, educational purposes.

If you know anyone keen on hard number sequences, here is a link to a few. It is a number sequence test. The above sequence is part of the test...

Number Sequence Test - Hard

6 thoughts on “A number sequence

  1. Mitchell (4KJ)

    Dear Ross,

    WOW, what a great post. My name is Mitchell and i’m in 4KJ this year. I figured out the first number pattern, it wasn’t too hard or too easy for me it was just right.

    The pattern was 0+1=1+1=2+3=5 etc.

    Kind Regards,
    Mitchell (4KJ)

    1. rossmannell

      Post author

      Dear Mitchell,

      I think it’s wonderful you have worked out how to solve the Fibonacci Sequence. It’s one of those patterns people find hard until they understand and then it seems easy. Well done. 🙂

      I would be very surprised if someone in your class had been able to work out the hard number sequence in this post. That one takes much more mathematical skill to solve.

      Ross Mannell

  2. Anthony

    To Ross Mannell,

    I don’t know how you did that. I just guessed and my guess was (c) 416.

    So my strategy was guess check and improve.

    did you have have a strategy and what was it?

    Anthony 😎

    1. rossmannell

      Post author

      Hello Anthony,

      Guess, check and improve is a good strategy. Knowing the real answer is (b) 400, we can try to work out how.

      My strategy for finding this answer was observation, number logic and a little trial and error (like guessing and checking). The first thing I noticed (observed) in the hard pattern was the numbers were close to double each step so I wrote down the numbers…
      15, 30, 60, 120, 240
      I then looked at the difference between the real numbers and the doubles. I noticed the following pattern…
      0, -1, -4, -12, -32
      Now I used my number logic skills and realised to get these numbers it must involve multiplying. A little trial and error and I saw the pattern I needed. Using this pattern, I knew the next number had to be 400.

      The hard sequence is not one I think primary school students could solve without help. Maybe some high school students or adults will solve it because they have more number experience.

      You would have to look very carefully at the pattern I gave to work out what number came after 400. I can give you a clue, the next number after 400 is higher than 700 but less than 1000.

      Ross Mannell

  3. Soha Farhin Pine

    Great post!

    I’m a 7th grader, and I did understand how you found the next number of this series.

    I had solved the series by myself, before I say your solution. It’s somewhat different.

    My initial observation was similar. Each number in the series is roughly double its predecessor.

    Then, I used trial-and-error to find out this pattern:


    The problem was quite interesting.


Leave a Reply

Your email address will not be published.