Each number in the sequence is calpengarahan a term (or sometimtape "element" or "member"), read Sequencpita pengukur and Seritape for a more in-depth discussion.

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## Finding Missingi Numbers

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

Sometimpita pengukur we can hanya 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, ...

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

x25 = 252 = 625

### 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 apa melakukan xn-1 mean? It meapejarakan "the previous term" as term mageri n-1 is 1 less than term mageri n.

And xn-2 meamenjadi the term sebelum 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 troubltape with findinew york "the lanjut number" in a sequence is that mathematics is so powerful we can find more than one Rule that works.

### maafkan saya is the next mageri in the sequence 1, 2, 4, 7, ?

di sini are three solutiopagi (tdi sini can be more!):

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 berbeda sequences.

Which is right? They are all right.

And tdi sini are other solutiomenjadi ... ... It may be a list of the winners" numbers ... So the lanjut number bisa be ... Anything!

## Simplest Rule

When in doubt choose the simplest rule that makpita pengukur sense, but tambahan mention that tdi sini are other solutions.

## Findingai Differences

Sometimtape 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:

n: 1 2 3 4 5 kapak (xn): 2n: Wrongai by:
7 9 11 13 15
2 4 6 8 10
5 5 5 5 5

The terakhir row shows that we are alcara wrong by 5, so hanya add 5 and we are done:

Rule: xn = 2n + 5

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

## Second Differences

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

... And then find the differencpita of those (called second differences), liusai 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 hanya n22:

n: 1 2 3 4 5 kapak (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, but seem to be drifting by 0.5, so let us try: n22n2

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

Wrongai by 1 now, so let us add 1:

n22n2 + 1 Wrong by:
 1 2 4 7 11 0 0 0 0 0

We did it!

The formula n22n2 + 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, ...

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