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Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

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Squaresearch

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

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Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Three Frogs

Stage: 4 Challenge Level: Challenge Level:1

3 Funny coloured frogs

Three frogs came in through the kitchen door

Said one to the others "let's explore"

The frogs hopped in and onto the table.

They sat in a line and admired the view - the red frog on the left the green in the middle and the blue frog on the right.

In their excitement they hopped over each other - one frog hopping randomly over any adjacent frog.

After 999 hops they stopped!

Why I do not know - perhaps they were exhausted.

Is it possible, after their 999 hops, for them to have ended up in the position in which they started?