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Broad Topics > Sequences, Functions and Graphs > Sequences

Lower Bound

Stage: 3 Challenge Level:

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Stage: 3 Challenge Level:

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Intersecting Circles

Stage: 3 Challenge Level:

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Converging Means

Stage: 3 Challenge Level:

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Triangles Within Triangles

Stage: 4 Challenge Level:

Can you find a rule which connects consecutive triangular numbers?

Triangles Within Pentagons

Stage: 4 Challenge Level:

Show that all pentagonal numbers are one third of a triangular number.

1 Step 2 Step

Stage: 3 Challenge Level:

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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.

Farey Sequences

Stage: 3 Challenge Level:

There are lots of ideas to explore in these sequences of ordered fractions.

Stage: 3 Challenge Level:

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

Paving Paths

Stage: 3 Challenge Level:

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Sissa's Reward

Stage: 3 Challenge Level:

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

More Pebbles

Stage: 2 and 3 Challenge Level:

Have a go at this 3D extension to the Pebbles problem.

Odd Differences

Stage: 4 Challenge Level:

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Clock Squares

Stage: 3 Challenge Level:

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

Logoland - Sequences

Stage: 3 Challenge Level:

Make some intricate patterns in LOGO

Towers

Stage: 3 Challenge Level:

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

Triangles Within Squares

Stage: 4 Challenge Level:

Can you find a rule which relates triangular numbers to square numbers?

Loopy

Stage: 4 Challenge Level:

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?

Tiny Nines

Stage: 4 Challenge Level:

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Changing Places

Stage: 4 Challenge Level:

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

A Little Light Thinking

Stage: 4 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Three Frogs

Stage: 4 Challenge Level:

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

Odds, Evens and More Evens

Stage: 3 Challenge Level:

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Seven Squares

Stage: 3 and 4 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Shifting Times Tables

Stage: 3 Challenge Level:

Can you find a way to identify times tables after they have been shifted up?

Differs

Stage: 3 Challenge Level:

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .

Stage: 3 and 4 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Charlie's Delightful Machine

Stage: 3 and 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Extending Great Squares

Stage: 2 and 3 Challenge Level:

Explore one of these five pictures.

Summing Squares

Stage: 4 Challenge Level:

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

What an Odd Fact(or)

Stage: 3 Challenge Level:

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Triangular Triples

Stage: 3 Challenge Level:

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

A Number Sequences Resource

Stage: 4 Challenge Level:

This resource contains interactive problems to support work on number sequences at Key Stage 4.

Sixational

Stage: 4 and 5 Challenge Level:

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. . . .

Remainder

Stage: 3 Challenge Level:

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Dalmatians

Stage: 4 and 5 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

LOGO Challenge - Sequences and Pentagrams

Stage: 3, 4 and 5 Challenge Level:

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

Triangles and Petals

Stage: 4 Challenge Level:

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Generating Number Patterns: an Email Conversation

Stage: 2, 3 and 4

This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.

Series Sums

Stage: 4 Challenge Level:

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

Pebbles

Stage: 2 and 3 Challenge Level:

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Squaresearch

Stage: 4 Challenge Level:

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

Ordered Sums

Stage: 4 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

Big Powers

Stage: 3 and 4 Challenge Level:

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Stretching Fractions

Stage: 4 Challenge Level:

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

Sums of Powers - A Festive Story

Stage: 3 and 4

A story for students about adding powers of integers - with a festive twist.

Two Much

Stage: 3 Challenge Level:

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.