We have two responses for you

Hi Eleanor,

A strategy that often works with sequences is to look at the differences between consecutive terms. Here, the differences are 6, 10, 15, and 21. These are special numbers because 6 = 4x3/2, 10=5x4/2, 15=6x5/2 and 21=7x6/2. This suggests to me that the difference between the 8th and 7th terms should be 8x7/2, and so on.

If this is right, then it should be possible to fill in the first and second terms of the sequence too. The 3rd term minus the 2nd term should be 3x2/2 =3, which makes the second term 1. For a similar reason, the first term should equal 0.

Now your sequence is 0, 1, 4, 10, 20, 35, 56, ...
Using the differences we figured out above:

1st term = 0
2nd term = 0 + 2x1/2 = 1
3rd term = 0 + 2x1/2 + 3x2/2 = (1x2 + 2x3)/2 = 4
4th term = 0 + 2x1/2 + 3x2/2 4x3/2= (1x2 + 2x3 + 3x4)/2 = 10

You can fill in the rest. Good luck!

Victoria

Hi Eleanor. Take a look at the gaps between the terms in the sequence:
10-4 = 6, 20-10 = 10, 35-20 = 15, 56-35 = 21.
So the "sequence of gaps" is 6, 10, 15, 21. Does this look familiar?
If not, let's look at the sequence of gaps between these terms.
10-6 = 4, 15-10 = 5, 21-15 = 6
So the second sequence of gaps is 4, 5, 6.

What do you think the two terms are in this sequence before the 4?

What does that make the two preceding terms in the 6, 10, 15, 21 sequence?

What does that make the two preceding terms in your original sequence?

Cheers,
Stephen La Rocque.