3 4, 7, 7, 13 pattern

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In following question, a number series is given with one term missing. Choose the correct alternative that will same pattern and fill in the blank spaces.

3, 4, 7, 7, 13, 13, 21, 22, 31, 44,

Option:

A. 42

B. 43

C. 51

D. 52

Answer: B . 43

Justification:

The given sequence is a combination of two series :
I.    Odd numbered terms i.e. 3, 7, 13, 21, 31, (....)
II.   Even numbered terms i.e. 4, 7, 13, 22, 44.
The pattern in I is + 4, + 6, + 8, + 10,....
The pattern in II is + 3, + 6, + 9, + 12,.....
.'. Missing term = 31 + 12 = 43.

Solution:

The given sequence of numbers is

3, 4, 7, 13, 14, 17, 23,…

We know that

4 - 3 = 1

7 - 4 = 3

13 - 7 = 6

17 - 14 = 3

23 - 17 = 6

The difference in the pattern 1, 3, 6 ….

Adding 1, 3 and 6 is repeated

So the next number should be 23 + 1 = 24

Where 24 - 23 = 1

Therefore, the next number should be 24.

Read the numbers and decide what the next number should be. 3, 4, 7, 13, 14, 17, 23,…?

Summary:

The next number in 3, 4, 7, 13, 14, 17, 23,… should be 24.

To find a missing number in a Sequence, first we must have a Rule

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for a more in-depth discussion.

Finding Missing Numbers

To find a missing number, first find a Rule behind the Sequence.

Sometimes we can just look at the numbers and see a pattern:

Example: 1, 4, 9, 16, ?

Answer: they are Squares (12=1, 22=4, 32=9, 42=16, ...)

Rule: xn = n2

Sequence: 1, 4, 9, 16, 25, 36, 49, ...

Did you see how we wrote that rule using "x" and "n" ?

xn means "term number n", so term 3 is written x3

And we can calculate term 3 using:

x3 = 32 = 9

We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" 25 wherever n is.

x25 = 252 = 625

How about another example:

Example: 3, 5, 8, 13, 21, ?

After 3 and 5 all the rest are the sum of the two numbers before,

That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:

3, 5, 8, 13, 21, 34, 55, 89, ...

Which has this Rule:

Rule: xn = xn-1 + xn-2

Now what does xn-1 mean? It means "the previous term" as term number n-1 is 1 less than term number n.

And xn-2 means the term before that one.

Let's try that Rule for the 6th term:

x6 = x6-1 + x6-2

x6 = x5 + x4

So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:

x6 = 21 + 13 = 34

Many Rules

One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works.

What is the next number in the sequence 1, 2, 4, 7, ?

Here are three solutions (there can be more!):

Solution 1: Add 1, then add 2, 3, 4, ...

So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc...

Rule: xn = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, ...

(That rule looks a bit complicated, but it works)

Solution 2: After 1 and 2, add the two previous numbers, plus 1:

Rule: xn = xn-1 + xn-2 + 1

Sequence: 1, 2, 4, 7, 12, 20, 33, ...

Solution 3: After 1, 2 and 4, add the three previous numbers

Rule: xn = xn-1 + xn-2 + xn-3

Sequence: 1, 2, 4, 7, 13, 24, 44, ...

So, we have three perfectly reasonable solutions, and they create totally different sequences.

Which is right? They are all right.

And there are other solutions ...

... it may be a list of the winners' numbers ... so the next number could be ... anything!

Simplest Rule

When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.

Finding Differences

Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.

Here is a simple case:

The differences are always 2, so we can guess that "2n" is part of the answer.

Let us try 2n:

The last row shows that we are always wrong by 5, so just add 5 and we are done:

Rule: xn = 2n + 5

OK, we could have worked out "2n+5" by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.

Second Differences

In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...

... and then find the differences of those (called second differences), like this:

The second differences in this case are 1.

With second differences we multiply by n22

In our case the difference is 1, so let us try just n22:

We are close, but seem to be drifting by 0.5, so let us try: n22 − n2

Wrong by 1 now, so let us add 1:

We did it!

The formula n22 − n2 + 1 can be simplified to n(n-1)/2 + 1

So by "trial-and-error" we discovered a rule that works:

Rule: xn = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...

Other Types of Sequences

Read Sequences and Series to learn about:

• Arithmetic Sequences
• Geometric Sequences
• Fibonacci Sequence
• Triangular Sequence

And there are also:

• Prime Numbers
• Factorial Numbers

And many more!

In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.

What would be the next number in the following series 4 7 13?

What is the number pattern for 3 6 12?

Which term of the series is 1280?

What are the next two numbers of this pattern 23 27 3135?