There are **43** NRICH Mathematical resources connected to **Arithmetic sequences**, you may find related items under Patterns, Sequences and Structure.

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

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

Can you complete this jigsaw of the multiplication square?

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?

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

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?

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

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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

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

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 3 dominoes?

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?

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?

Which of these pocket money systems would you rather have?

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

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

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

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

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

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

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

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

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?

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

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

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

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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?

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?

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

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?

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

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. . . .

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.

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?

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

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.

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?

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

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