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.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?
$$\frac{1}{2} + \frac{1}{4} $$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$
Work out the answers to Charlie's sums. What do you notice?
Will the pattern continue? How do you know?
Try writing an expression for $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}$$
Could you convince someone else that your expression is correct for all values of $n$?
Charlie drew a diagram to try to explain what was going on:
Use Charlie's diagram to explain why $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n} = 1-\frac{1}{2^n} = \frac{2^n-1}{2^n}$$
$$1 + 2$$ $$1 + 2 + 4$$ $$1 + 2 + 4 + 8$$
Work out the answers to Alison's sums. What do you notice?
Will the pattern continue?
How do you know?
Try writing an expression for $$1 + 2 + 4 + \dots + 2^n$$
Alison drew a diagram to try to explain what was going on:
Can you use Alison's diagram to explain why $$1 + 2 + 4 + \dots + 2^n = 2^{n+1}-1$$