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Tips on solving sequence puzzles logically

Sequence puzzles are commonly used in tests and quizzes. Sometimes they may be related to words and knowledge. This article looks purely at maths related puzzles where you need to find the mathematical pattern between members of the sequence, and thereby work out what the next term in the puzzle is.

As this works through, a couple of simple but effective techniques are explored for helping to discover the pattern between different elements of the sequence.

Here's a very simple sequence:

1,2,3,4,5

Instantly you know that '6' comes next.

But lets look at something very similar:

154,162,170,178,186

It may take a little longer to spot, but here we see that the next number is '194'. So how do we solve these puzzles quickly?

Well, this is a key technique for analysing sequences: work out the difference between different terms of the sequence, thus here the differences are +8, +8, +8, +8 and therefore we instantly find out what the sequence is.

When there is a constant addition or subtraction, we know the next term is n + 8, where 'n' is here the last item in the sequence we know the value of. So it is always useful to check the difference between terms. Let's do this again:

3, 9, 27, 81

Here the difference path looks like this: +6, +18, +54 (which in turn is +12, +36). What we notice here is that the difference between terms is not constant, but is instead increasing. This means we are going to be performing some sort of multiplication, so it won't be n+ or n- a figure, but rather xn, where x is a multiplying factor. To find the multiplication factor, we now divide the terms by each other to see that 54 / 18 and 18 / 3 is 3, so the multiplication factor is 3 each time, giving us that the next term is 81 x 3 or 243.

So seeing what is going on with the differences can help to solve these sums.

Remember that if you get alternating negative and positive terms in a sequence, you should consider that there is a negative multiplication factor being applied, e.g.

2, -4, 8, -16, 32 is a sequence being multiplied by -2n each time.

How about sequences that are not steadily increasing by addition, subtraction, multiplication or division?

For instance look at this one:

3, 6, 18, 72, 360

This time the differences show us this: 3, 12, 54, 288. Well, what we do know is there is a large increase each time, so there must be multiplication involved. However, it's not steady, as it increases. By dividing the terms we soon see that we are multiplying by 2 then 3 then 4 then 5... problem solved!

If you like this sort of game, then try these sequence puzzles
Maths Sequence Puzzles
Author: Dan

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Last Updated: Nov 1st 2007

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Questions Dan has Answered:

HI Dan, which number logically ends this sequence? 100 500 1 50 1000 5 ? Many thanks

So the sequence is: 100 500 1 50 1000 5 Multiply by 5, divide by 500 Multiply by 50, multiply by 20, divide by 200. No idea!

What is the next term in the following 1) 6,8,5,10,3,14,1 2) B,0,C,2,D,0,E,3,F,3,G 30 2,3,6,1,8,6,8,4,8,4,8,3,2,3,2,3

1) 6,8,5,10,3,14,1 +2, -3, + 5, -7, + 11, - 13 So looks like this alternates adding then subtracting primary numbers. Next primary is 17, so therefore add 17 to get the next term which is 18. 2) B,0,C,2,D,0,E,3,F,3,G 30 Alternating letters of the alphabets: B, C, D etc so the next term is H.

Triangle Sequences. 1 Triangle = 3 Match sticks 2 Triangles = 5 Match Sticks 3 Triangles = 7 Match Sticks. 1509 Triangles = How many Match Sticks

1 Triangle = 3 Match sticks 2 Triangles = 5 Match Sticks 3 Triangles = 7 Match Sticks. 1509 Triangles = How many Match Sticks Ok, so the pattern is 2n + 1. Therefore this (1509 x2) + 1. This is 3019.

What is the next number in this sequence: 1,18,4,13,6,10

So the sequence is: 1,18,4,13,6,10 There are many possible answers to this sequence so no absolute answer. One option is that this is two sequences interlinked, so the relevant one here is 1, 4, 6 ? which is add three, add two, then add one so the answer is 7. There are many other possible answers.

Hi, if a sequence is 3, 3, 5, 4, 4, 3, 5 how was this worked out and what is the next number?

3, 3, 5, 4, 4, 3, 5 is the sequence. Add 0, add 2, minus one, add zero, minus one, add 2. No obvious sequence here so not sure.

Complete the missing numbers - - - 112 - 162 - - 237

112 162 237 Add 50, add 75, so next add 100 to get 337.

Whqt is the sequence? 1,2,4,7,11,16

1,2,4,7,11,16 Add one, two, three, four, five so add six to get 22.

1 8 27 64

1,8,27,64. This is the cube of each number, e.g. 1x1x1, 2x2x2 etc so here we have 5x5x5 which is 125 as the next term.

Hi dan what no ends this sequence 1 8 27 64

125

Find the next 3 numbers in the following sequence: 2,3,5,7,11,13,17

2,3,5,7,11,13,17 This looks like a list of prime numbers, so next comes 19 then 23.

I dont understand how to work out n-3 from 1,5,6,11,17,24,32,41,51,62 please could you help from zak

6,11,17,24,32,41,51,62 Here you are adding 5 then 6 then 7 then 8 then 9,10,11 so next 12 to get 74.

Whose famous sequence starts1123581321

Yes this is called the fibbonaci sequence.

3, 6, 18, 72, 360 what is the next number

Multiply by 2, then 3, then 4 and so on. So here we have 360 x 6 so it is 2160.

Hi could you please tell me the next two numbers in this sequence 0,2,3,6,7,1,_,_,

So the sequence is 0,2,3,6,7,1,_,_, What comes next is 9 then 4 then 5 then 8. It is the sequence of numbers written as words in english in reverse alphabetical order - quite a nasty sequence to work out as it's not a mathematical relationship.

What are the next two numbers in this sequence? 2, 5, 11

2, 5, 11... there are only three terms here so there are many possible sequences. However, the most obvious is to double and add one each time. This makes 11x2 + 2 the next term: 24, followed by 50. In sum then, this is 2,5,11,24,50.

To find a missing number in a pattern, you may have to try more than one ____________.

... possible combination. This is basically saying a trial and error method could work; however there is more than one possible way to finish this sentence - in fact several answers, so your teacher must be looking for a sensible answer not a specific one!

What is next in the sequence: 1,4,?,?,?,36,?

1,4,?,?,?,36,? This is the square numbers, e.g. 1x1, 2x2. So we have 1,4,9,16,25,36,49

I have to give the next three answers to this sequence 2 3 5 7 11 13 17

So we have... 2 3 5 7 11 13 17 This is the sequence of prime numbers so we have 19 then 23 next.

Write a number in the brackets so they are in order -8,( ),-5,( )-2( )

There are lots of choices - it could be -7 or -6 in the first bracket, then -4 or -3 in the second bracket, then any number from -1 up to positive infinity in the third bracket

What are the next terms and the formula for the sequence : 3;6;7;-2 ?

Can't see anything obvious here.

Hi.please can you tell me what comes next in the following 3 thanks 0 2 3 6 7 1 - - 20 1 18 4 13 - - 8 5 4 9 1 - -

0 2 3 6 7 1 9 then 4 20 1 18 4 13 6 then 10 8 5 4 9 1 7 then 6

What is the sequence 15 19 _ _ _ 65 84

Not sure on this one.

What is the next number in the sequence 2,4,3,9,8,-?

Sequence appears to be 'square it, subtract one, square it, subtract one' therefore next number is 64.

Hi can you help with the next 2 numbers in the sequence 20 1 18 4 13

Several possible sequences here depending whether you look for one or two interweaving sequences etc. One possible answer is 6,10.

Hi Dan, Could you help me work out this sequence please 2,6,8,14,22,36

Add 22 to get 58 - basically add the previous term each time, e.g. to get 8 add 2 to 6, to get 14 add 6 to 8 and s on.

Wthat is the rule for this sequence. 0, 1, 4, 11

0, 1, 4, 11: Add 1, add 3, add 7. So the sequence could be 'double the difference and add one' meaning the next term would be 11 + 15 = 26.

3354435 what is the next two numbers thanks

This is one long number... not sure how you can work out what comes next :s

What are the 50th, 63rd and 100th numbers in the sequence below? 4, 5, 8, 8, 9, 9, 12, 13, 13, 13, 17, 18, ... How do you do this?

4, 4, 5, 8, 8, 9, 9, 12, 13, 13, 13, 17, 18, 21, 22, 22, 23, 26, 26, 27, 26, 30, 31, 34, 34, 35, 35, 38, 39, 39, 36, 40, 41, 44, 44, 45, 45, 48, 49, 49, 45

Hello dan, can you tell me a formula for this sequence 2 4 4 6 6 8 8 10 8 8 6 6 4 4 2, and how you found it?

Looks like 'add two, add nothing, add two, add nothing etc' then start taking away from 10.

What is the connecting rule here (S) 1 2 3 4 5 (t) 1 3 7 15 31

Double and add one each time, so for the next term double and add one to get 63.

What are the following pattern to 2,3,5,7,11,13? please tell me no later then 9

This looks like the sequence of odd numbers.

HI Dan, Have a few sequences I need help with, need to find next 2 numbers in each sequence. 7-0-1-2-7-1-2-3 20-23-23-23-25-25-26-27-27 87-69-75-61-63-53-51 36-42-35-28-45-21-14-7 Thanks Pete

7-0-1-2-7-1-2-3 Looks like it might be 7 next... not sure on the others though!

Help 1 11 21 1211 111221 312211 13112221 1113213211 Next?

1113213211

FIND RHE MISSING NUMBER 1 3 4 13 ?

FIND THE MISSING NUMBER 1 3 4 13 ? Several options. One possible sequence is to add on the square of the previous number; so 4 is 3 + 1x1, 13 is 4 + 3x3 so next would be 13 + 4x4 = 29

What numbber comes next and how do you know 1,4,9,1,6,2,5,3,6,4,9,6,4

8,1 comes next. It is the squares of each number in turn with commas inbetween. e.g. 1 squared, 2 squared, 3 squared, 4 squared etc is 1,4,9,16 but with commas added between every digit becomes 1,4,9,1,6...

1.....11.....21.....1,211.....111,221.....312,211 what would be the next # and explain

13112221 comes next. This is one of those rather silly sequences that is not mathematical but descriptive. Start with one, then describe it - there is one one, so that's 11. Next describes this, there are two ones, or 21, and so on.

How do you finish this sequence 0,2,3,6,7,1

9,4,5,8

I have the sequence 9, a, 4, b. If b=8/3, how do you get that answer and what is a?

9, a, 4, 8/3 The answer is 6. The sequence is dividing by 1.5 for each term.

Which best completes the sequence 3,4,6,9,13

18 comes next. This is add one, then add two, then add three etc, so we add five to get 18.

Dan, help.... trying to do my son's homework and stuck on this one... 2, 2, 3, 4, 6, 9, 14, ___, ___, ___ help

2, 2, 3, 4, 6, 9, 14, The next number is 22; it looks like we sum the differences between the previous two terms to find the next item in the sequence: 0,1,1,2,3,5 - so since 5+3 is 8 then the next number is 14 + 8 = 22.

Please solve What are the next terms in the sequence 25,36,47?

25,36,47... Looks like this is adding 11 each time, so next up will be 58.

1,1,2,3,5,8,13,21

1,1,2,3,5,8,13,21 Add the previous two terms each time, so next is 21 + 13 = 34

Whats next in the sequence? 3, 8, 17, 26, 34,....

See previous answer!

How do you do 1,1,2,3,5,8,13,21...... pretty pretty please with a cherry on top answer my question

Add the previous two numbers; next is 34.

What is the next number in the number pattern 2 10 12 17 18 19

200, these numbers start with 'D' in French

What is the rule in this sequence 1 , 4 , 14

Add three, add ten, so looks like adding seven more each time, so now add 17, so 31.

Dan : I have a number sequence for you to solve? I am wondering if it is a valid sequence or not? Here it is as follows? 1,4,10,19,31,__. Possible answers are:a(36), b(48),c (46), or no apperent pattern is answer d? Dale

Add 3, 6, 9, 12 so next add fifteen making the answer c - 46.

What is the next number in this sequence 2, 18, 1, 9, 14, 16, 15, 23, 5,

6

What is the nth term of this sequence? 4,7,12,19,28

Add 3, 5, 7, 9 so next add 11 to get 39. So n = n-2 + (n-2 - n-1) + 2

4 3 6 3 2 7 2 1 8 1 .. What is the next term in this sequence?

0, 9

64,16,4,1,1/4 which comes next

Divide by four each time, next in this maths sequence is therefore 1/16

Hi Dan, Can you explain how you get 6 & 10 as the next two numbers in the sequence 20, 1, 18, 4, 13? Many thanks.

20, 1, 18, 4, 13, 6, 10. No idea sorry!

The sequence 3,3,5,4,4,3,5 is explained thus: the nth terms of the sequence, 1,2,3,4,5,6,7 are spelt one two three four five six seven the sequence is thus the number of letters in each word.

Good answer.

What is the next number in tis sequence: 1 2 4 3 6 8 7 14 16 ?

Take away one, double it, add two in turn, so 15 next.

1, 2, 6, 42, 1806 Hi Dan can you find the next number in the above sequence please?

3263442 All you do is multiply by the previous number plus 1, eg 2 is (1+1)x1 6 is (2+1)x2 42 is (6+1)x6

2n+6=0

Re-arrange for n is it? Well 2n = -6 Therefore n = -3

Hi Dan I\'m stumped! Can you complete this sequence please? ...,7.5,3,1.2,...,0.192,0.0768

It is 0.48 - just multply by 2.5 at each stage of the sequence.

Childs Homework im stuck the sequence is CYGTNLIT

The answer is 'S': can you get the next letter in this Sequence.

How do you find a formula for finding the nth term in a square number sequence?

The formula is just n x n For instance in the sequence of squares: 1, 4, 9, 16 this is 1 x 1, 2 x 2, 3 x 3, 4 x 4 - so therefore the nth term is just n x n

Dan, we know the numerical answer to a sequencial problem but we do not know how to write it out. (1,6,12,19). we know the answer is 27, but how does he state it in words.

Add 5, then 6, then 7, then 8. So the nth term is (n-1) + (i+3) where n is the given term in the sequence and i is the position of the term. So to get six for instance we get the n - 1 term which is 1. Then we add to this the position of six in the sequence (position two) and add three to that, so 1 + 5 = 6

What are the next three numbers in the pattern 1 11 21 1211 111221 312211 13112221?

1113213211 comes next

0,2,5,10,17,28,41,58,__,__,__ find the pattern and the next three numbers in the sequence

Add 2,3,5,7,9...

Hi. i am stuck on a question. what type of numbers are 2 4 8 16 32 and 64. thanks

These are even numbers - all multiples of 2!

Find the missing term in the sequence? Q.1. 7 9 5 11 4 15 12 7 13 8 11 ? --------------------------------------- Q.2. 1 3 4 8 15 ? --------------------------------------- Q.3. 58 35 ? 11 6 3 ---------------------------------------

Q2: various options, could be sum the previous three numbers to get the next number, if so 15+8+4 = 27.

What is the missing number (and explanation) for this sequence? 12 15 19 21 22 22 24 26 26 ?? 29

Don't know

What is the rest of the pattern 1 5 13 29?

Double it and add three, eg 2 x 29 + 3 = 61

What comes next in these 2 sequences please. 4 5 4 2 5 3 ? 20 20 8 10 15 ? thanks sc

Not sure on this one... looking at all these sequence questions on this page is like a full time job!!! ;)

Hey Dan, I just took a test with random sequence questions, and I think I remembered the numbers from one that really stumped me: 123,615,1002,___ I\'m stumped! And, for some reason, I *need* to know the answer. Any help?

With only three terms there are many possible sequences that fit. The differences between these terms are: 497, 387 so one sequence could be add 497 then 110 less each time so next you would add 277 to get 1279. But with so few terms there are many possible answers so they should really have asked you a sequence with more terms!

What is the formula for this sequence: 0, 100, 300, 700, 1400, 2500, 4100, 6300, 9200, 12900

Add 200, 400, 700, 1100, 1600, 2200, 2900

What is the missing number in this sequence 1, 8, 27, ?, 125, 216. and how do we find it?

This is the sequence of cubes, 4 x 4 x 4 = 64

1, 2, 5, 10, 20, next two numbers and why please

Looks like 50,100 as the coins in legal tender in the UK in pence.

What are the next 2 numbers in the sequence 12,5,20,1,18,14

Not sure.

1, 2, 6, 42, 1806 ?

Answer is 3263442. The sequence is the number x the number + 1, therefore next is (1806 x 1807).

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