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Can you see how this picture illustrates the formula for the sum of the first six cube numbers? ### Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures. ### Series Sums

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

# Difference Sequence

##### Age 14 to 16 Short Challenge Level:

The first term in a sequence is 2016 and the second term is 2017.

Every other term is given by the difference between the two terms before it. So for example, the third term is the difference between 2016 and 2017, which is 1.

The $n^\text{th}$ term is equal to 2000; what is the value of $n$?

This problem is taken from the World Mathematics Championships
You can find more short problems, arranged by curriculum topic, in our short problems collection.