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Broad Topics > Sequences, Functions and Graphs > Limits of Sequences

Litov's Mean Value Theorem

Stage: 3 Challenge Level:

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Squareness

Stage: 5 Challenge Level:

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Slide

Stage: 5 Challenge Level:

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

Approximating Pi

Stage: 4 and 5 Challenge Level:

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Climbing Powers

Stage: 5 Challenge Level:

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or. . . .

Summing Geometric Progressions

Stage: 4 and 5 Challenge Level:

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

Stage: 5 Challenge Level:

Compares the size of functions f(n) for large values of n.

Light Blue - Dark Blue

Stage: 2 Challenge Level:

Investigate the successive areas of light blue in these diagrams.

Ruler

Stage: 5 Challenge Level:

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

Small Steps

Stage: 5 Challenge Level:

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Infinite Continued Fractions

Stage: 5

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

Continued Fractions II

Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Continued Fractions I

Stage: 4 and 5

An article introducing continued fractions with some simple puzzles for the reader.

Sums and Products of Digits and SP Numbers

Stage: 5

This article explores the search for SP numbers, finding the few that exist and the proof that there are no more.

Try to Win

Stage: 5

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

Zooming in on the Squares

Stage: 2 and 3

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Archimedes and Numerical Roots

Stage: 4 Challenge Level:

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Little and Large

Stage: 5 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Happy Numbers

Stage: 3 Challenge Level:

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.