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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.?

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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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LOGO Challenge - Sequences and Pentagrams

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Weekly Challenge 34: Googol

Age 16 to 18 Short Challenge Level:

A googol is the number $10^{100}$. What is the smallest whole number $n$ for which
n^4-6n^2> \mbox{googol}?
Can you carefully write out the number on the left-hand side of this inequality for this value of $n$ in base 10?

Did you know ... ?

Although computers are very useful in checking calculations in number theory, it is very difficult to use them to perform calculations involving very, very large numbers. A googol is a big number, but a googolplex, defined to be $10^{\mbox{googol}}$ is unimaginably larger. Such gigantic numbers do make appearances in mathematics from time to time and require the power of pure thought and mathematics to yield to analysis. Rather amusingly, the Googol Corporation call their headquarters 'the googolplex'.