# 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

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 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, ...Xem thêm: Ánh Sáng Truyền Đi Trong Chân Không Với Vận Tốc Gần Bằng Bao Nhiêu ?**

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

Which is right? **They are all right.**

... 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 *n2***2**

In our case the difference is 1, so let us try just *n2***2**:

n: 1 2 3 4 5

**Terms (xn):**

*n2*

**2**: 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: *n2***2** − *n***2**

*n2*

**2**−

*n*

**2**Wrong by:

0 | 1 | 3 | 6 | 10 |

1 | 1 | 1 | 1 | 1 |

Wrong by 1 now, so let us add 1:

*n2*

**2**−

*n*

**2**+ 1 Wrong by:

1 | 2 | 4 | 7 | 11 |

0 | 0 | 0 | 0 | 0 |

We did it!

The formula ** n22 − n2 + 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, ...Xem thêm: Ielts Writing: Advantages & Disadvantages Of Living In The City**

## Other Types of Sequences

Read Sequences và Series to learn about:

And there are also:

And many more!

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