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

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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?

Double Trouble

Age 14 to 16 Challenge Level:

Why do this problem

In this problem a surprising number pattern can be explained by using an image. By understanding the significance of the image offered students are helped to perceive the general rule.


Possible approach

This printable worksheet may be useful: Double Trouble.

"Imagine a sequence of fractions where each one is half of the previous fraction."

Write these sums on the board and ask students to work them out:

$$\frac{1}{2} + \frac{1}{4} $$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$

"What do you notice?"
"Do you think the pattern will continue?"
"How do you know?"

Offer students a chance to share their ideas, and then show the video, or recreate Charlie's diagram on the board. Perhaps ask students to recreate the diagram for themselves.

"How could you use the diagram to explain the patterns you have noticed?"
"Can you describe an expression for the sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\dots + \frac{1}{2^n}$?"
"Can you convince someone that your expression is correct for all values of $n$?"

Allow students some time to discuss in pairs, then bring the class together to share their insights. It is important to insist on clearly justified arguments that refer to the generality - a key question to ask is "How do you know it will always happen?".
The second part of the problem, looking at the sum of the sequence $1 + 2 + 4 + \dots + 2^n$, can be treated in the same way.

Possible extension

Diminishing Returns uses similar diagrams to explore other geometric series and can lead on to discussion of infinite sums.

Possible support

Students could start by working on Slick Summing, in which they are invited to explore the sums of simple arithmetic sequences.