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Route to Root
A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?
Squaresearch
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?
Stage: 4 and 5 Challenge Level: 
The $n$th 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 $6$.
NRICH is part of the family of activities in the Millennium Mathematics Project.