Each number in the succession is referred to as a term (or sometimes "element" or "member"), check out Sequences and collection for a an ext in-depth discussion.

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## Finding lacking Numbers

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

Sometimes we deserve to just look in ~ the numbers and also 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, ...

We deserve to use a preeminence to find any type of term. Because that example, the 25th term can be uncovered by "plugging in" 25 where n is.

x25 = 252 = 625

How around another example:

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

After 3 and also 5 every the rest are the sum the the two numbers before,

That is 3 + 5 = 8, 5 + 8 = 13 etc, i beg your pardon 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 walk xn-1 mean? It means "the previous term" together term number n-1 is 1 less than hatchet number n.

And xn-2 method the term before that one.

Let"s shot that preeminence for the 6th term:

x6 = x6-1 + x6-2

x6 = x5 + x4

So term 6 equates to term 5 plus ax 4. We currently know term 5 is 21 and term 4 is 13, so:

x6 = 21 + 13 = 34

## Many Rules

One that the troubles v finding "the following number" in a sequence is that mathematics is so an effective we deserve to find an ext than one dominance that works.

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

Here room three options (there can be more!):

Solution 1: include 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 preeminence looks a little complicated, yet it works)

Solution 2: after 1 and 2, add the two previous numbers, to add 1:

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

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

Solution 3: after 1, 2 and also 4, add the 3 previous numbers

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

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

So, we have actually three perfect reasonable solutions, and they create completely different sequences.

Which is right? They room all right.

And over there are various other solutions ... ... It may be a perform of the winners" numbers ... So the next number could be ... Anything!

## Simplest Rule

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

## Finding Differences

Sometimes it helps to uncover the differences in between each pair of numbers ... This can regularly reveal an basic pattern.

Here is a an easy case: The distinctions are always 2, for this reason we deserve to guess the "2n" is part of the answer.

Let us shot 2n:

n: 1 2 3 4 5 state (xn): 2n: not correct by:
7 9 11 13 15
2 4 6 8 10
5 5 5 5 5

The critical row mirrors that us are always wrong through 5, for this reason just include 5 and we space done:

Rule: xn = 2n + 5

OK, we could have worked out "2n+5" by simply playing around with the number a bit, but we desire a systematic means to perform it, for as soon as the sequences get much more complicated.

## Second Differences

In the succession 1, 2, 4, 7, 11, 16, 22, ... we require to uncover the distinctions ...

... And then uncover the differences of those (called second differences), choose this: The second differences in this situation are 1.

With second differences us multiply by n22

In our case the difference is 1, for this reason let us shot just n22:

n: 1 2 3 4 5 Terms (xn):n22: wrong by:
1 2 4 7 11
0.5 2 4.5 8 12.5
0.5 0 -0.5 -1 -1.5

We are close, however seem to it is in drifting through 0.5, so let united state try: n22n2

n22n2 dorn by:
 0 1 3 6 10 1 1 1 1 1

Wrong by 1 now, for this reason let us include 1:

n22n2 + 1 not correct by:
 1 2 4 7 11 0 0 0 0 0

We go it!

The formula n22n2 + 1 can be simplified to n(n-1)/2 + 1

So by "trial-and-error" we uncovered a ascendancy that works:

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

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

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