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There are **49** NRICH Mathematical resources connected to **Arithmetic sequences**, you may find related items under Patterns, sequences and structure.

Problem
Primary curriculum
Secondary curriculum
### Growing Surprises

Can you find the connections between linear and quadratic patterns?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make Now?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Squares

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Steel Cables

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odds, Evens and More Evens

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Age 11 to 14

Challenge Level

Problem
Live Primary curriculum
Secondary curriculum
### Charlie's Delightful Machine

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

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Little Light Thinking

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shifting Times Tables

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Slick Summing

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mystic Rose

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Elevenses

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Square Jigsaw

Can you complete this jigsaw of the multiplication square?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squares in Rectangles

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coordinate Patterns

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Squares - Group-worthy Task

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Number Pyramids

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Age 11 to 14

Challenge Level

Interactive
Primary curriculum
Secondary curriculum
### Proof Sorter - Sum of an Arithmetic Sequence

Put the steps of this proof in order to find the formula for the sum of an arithmetic sequence

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Prime AP

What can you say about the common difference of an AP where every term is prime?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Missing Middles

Can you work out the domino pieces which would go in the middle in each case to complete the pattern of these eight sets of three dominoes?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Buzzy Bee

Buzzy Bee was building a honeycomb. She decorated the honeycomb with a pattern using numbers. Can you discover Buzzy's pattern and fill in the empty cells for her?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Maxagon

What's the greatest number of sides a polygon on a dotty grid could have?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Speedy Summations

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Prime Sequences

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mobile Numbers

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Investigating Pascal's Triangle

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tables Without Tens

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unravelling Sequences

Can you describe what is happening as this program runs? Can you unpick the steps in the process?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squares, Squares and More Squares

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polite Numbers

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cherries Come in Twos

Susie took cherries out of a bowl by following a certain pattern. How many cherries had there been in the bowl to start with if she was left with 14 single ones?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Six in a Circle

If there is a ring of six chairs and thirty children must either sit on a chair or stand behind one, how many children will be behind each chair?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Transformations Tables

These grids are filled according to some rules - can you complete them?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Red Balloons, Blue Balloons

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Series Sums

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Natural Sum

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summats Clear

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Janusz Asked

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Skip Counting

Find the squares that Froggie skips onto to get to the pumpkin patch. She starts on 3 and finishes on 30, but she lands only on a square that has a number 3 more than the square she skips from.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Alphabet Blocks

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Great Tiling Count

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Matchsticks

Reasoning about the number of matches needed to build squares that share their sides.

Age 7 to 11

Challenge Level