There are **45** NRICH Mathematical resources connected to **Arithmetic sequences**, you may find related items under Patterns, Sequences and Structure.

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

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

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

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Can you complete this jigsaw of the multiplication square?

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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?

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

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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?

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

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

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

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

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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?

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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?

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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?

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

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

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

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

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

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

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

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

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

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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?

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

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

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

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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?

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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?

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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?

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

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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?

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

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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. . . .

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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.

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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?

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

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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.

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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?

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

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