Each number in the sequence is called a term (or occasionally "element" or "member"), check out Sequences and series for a much more in-depth discussion.

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## Finding missing Numbers

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

Sometimes we can just look in ~ 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, ...

We have the right to use a ascendancy to find any term. Because that example, the 25th term have the right to be discovered by "plugging in" 25 wherever n is.

x25 = 252 = 625

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

After 3 and also 5 all the remainder are the sum the the 2 numbers before,

That is 3 + 5 = 8, 5 + 8 = 13 etc, i beg your pardon is component of the Fibonacci Sequence:

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

Which has actually this Rule:

Rule: xn = xn-1 + xn-2

Now what walk xn-1 mean? It means "the vault term" together term number n-1 is 1 much less than term number n.

And xn-2 method the term before that one.

Let"s try that ascendancy for the 6th term:

x6 = x6-1 + x6-2

x6 = x5 + x4

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

x6 = 21 + 13 = 34

## Many Rules

One that the troubles with finding "the following number" in a sequence is that math is so powerful we have the right to find much more than one dominion that works.

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

Here room three services (there have the right to be more!):

Solution 1: include 1, then include 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 dominance looks a bit complicated, yet it works)

Solution 2: after ~ 1 and also 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: ~ 1, 2 and also 4, include the three previous numbers

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

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

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

Which is right? They room all right.

And over there are various other solutions ...
 ... It might be a list of the winners" numbers ... For this reason the next number might be ... Anything!

## Simplest Rule

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

## Finding Differences

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

Here is a an easy case:

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

Let us shot 2n:

n: 1 2 3 4 5 terms (xn): 2n: wrong 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 by 5, for this reason just add 5 and we space done:

Rule: xn = 2n + 5

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

## Second Differences

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

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

The second differences in this situation are 1.

With second differences we multiply by n22

In our case the distinction is 1, for this reason let us try just n22:

n: 1 2 3 4 5 Terms (xn):n22: not correct by:
1 2 4 7 11
0.5 2 4.5 8 12.5
0.5 0 -0.5 -1 -1.5

We room close, but seem to it is in drifting through 0.5, therefore let us try: n22n2

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

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

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

We go it!

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

So through "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, ...

See more: List The Elements Of Group 5A — The Pnictogens, The Other Group 5A Elements: P, As, Sb, An D Bi