# What Is The Nth Term For The Sequence 1, 4, 9, 16, 25?

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

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## Finding Missing Numbers

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

Sometimes we can just look at the numbers và 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 "plugging 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 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 và 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: địa chỉ cửa hàng 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 và 2, địa chỉ 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, địa chỉ cửa hàng the three previous numbers

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

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

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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 danh mục 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 lớn 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 Terms (xn): 2n: Wrong by:
7 9 11 13 15
2 4 6 8 10
5 5 5 5 5

The last row shows that we are always wrong by 5, so just showroom 5 & 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 vì 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 ...

... Và then find the differences of those (called second differences), lượt thích 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:

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, but seem to lớn be drifting by 0.5, so let us try: n22n2

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

Wrong 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 lớn 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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