What is the 9th term in the sequence of square numbers

A sequence is an ordered list of numbers. It can be thought of as a function, f(n), where the argument, n, takes on the natural-number values 1, 2, 3, 4,... (or occasionally 0, 1, 2, 3, 4,...). A sequence can follow a regular pattern or an arbitrary one. It may be possible to compute the value of f(n) with a formula, or it may not.

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Triangular Numbers
Square Numbers
Cube Numbers
Fibonacci Numbers
Other Sequences

The terms of a sequence are often represented by letters with subscripts, an, for example. In such a representation, the subscript n is the argument and tells where in the sequence the term an falls. When the individual terms are represented in this fashion, the entire sequence can be thought of as the set, or the set where n is a natural number. This set can have a finite number of elements, or an infinite number of elements, depending on the wishes of the person who is using it.

One particularly interesting and widely studied sequence is the Fibonacci sequence: 1, 1, 2, 3, 5, 8,.... It is usually defined recursively: an = an-2 + an-1. In a recursive definition, each term in the sequence is defined in terms of one or more of its predecessors (recursive definitions can also be called "iterative"). For example, a6 in this sequence is the sum of 3 and 5, which are the values of a4 and a5, respectively.

Another very common sequence is 1, 4, 9, 16, 25,..., the sequence of square numbers. This sequence can be defined with the simple formula an = n2, or it can be defined recursively: an = an-1 + 2n - 1.

Another sequence is the sequence of prime numbers: 2, 3, 5, 7, 11, 13,.... Mathematicians have searched for centuries for a formula which would generate this sequence, but no such formula has ever been found.

One mistake that is made frequently in working with sequences is to assume that a pattern that is apparent in the first few terms must continue in subsequent terms. For example, one might think from seeing the five terms 1, 3, 5, 7, 9 that the next term must be 11. It can, in fact, be any number whatsoever. The sequence can have been generated by some random process such as reading from a table of random digits, or it can have been generated by some obscure or complicated formula. For this reason a sequence is not really pinned down unless the generating principle is stated explicitly. (Psychologists who measure a subject's intelligence by asking him or her to figure out the next term in a sequence are really testing the subject's ability to read the psychologist's mind.) Sequences are used in a variety of ways. One example is to be seen in the divide-and-average method for computing square roots. In this method one finds the square root of N by computing a sequence of approximations with the formula an = (an-1 + N/an-1)/2. One can start the sequence using any value for a1 except zero (a negative value will find the negative root). For example, when N = 4 and a1 = 1

This example illustrates several features that are often encountered in using sequences. For one, it often only the last term in the sequence that matters. Second, the terms can converge to a single number. Third, the iterative process is one that is particularly suitable for a computer program. In fact, if one were programming a computer in BASIC, the recursive formula above would translate into a statement such as R = (R + N/R)/2.

Not all sequences converge in this way. In fact, this one does not when a negative value of N is used. Whether a convergent sequence is needed or not depends on the use to which it is put. If one is using a sequence defined recursively to compute a value of a particular number only a convergent sequence will do. For other uses a divergent sequence may be suitable.

Mortgage companies often provide their customers with a computer print-out showing the balance due after each regular payment. These balances are computed recursively with a formula such as An = (An-1)(1.0075) - P, where An stands for the balance due after the n-th payment. In the formula (An-1)(1.0075) computes the amount on a 9% mortgage after one month's interest has been added, and (An-1)(1.0075) - P the amount after the payment P has been credited. The sequence would start with A0, which would be the initial amount of the loan. On a 30-year mortgage the size of P would be chosen to bring A360 down to zero. This sequence converges, but very slowly for the first few years.

Tables, such as tables of logarithms, square roots, trigonometric functions, and the like are essentially paired sequences. In a table of square roots, for example the column on the left is a sequence and the column on the right the sequence where each bn = the square root of an. By juxtaposing these two sequences, one creates a handy way of finding square roots.

Sequences are closely allied with (and sometimes confused with) series. A sequence is a list of numbers; a series is a sum. For instance 1/1, 1/2, 1/3, 1/4,... is a harmonic sequence; while 1/1 + 1/2 + 1/3 + 1/4 +... is a harmonic series.

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An Arithmetic Sequence is made by adding the same value each time.

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:

The value added each time is called the "common difference"

What is the common difference in this example?

Answer: The common difference is 8

The common difference could also be negative:

25, 23, 21, 19, 17, 15, ...

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

1739, 1740, 2511, 9771, 9772

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A Geometric Sequence is made by multiplying by the same value each time.

This sequence has a factor of 3 between each number.
The pattern is continued by multiplying by 3 each time, like this:

What we multiply by each time is called the "common ratio".

In the previous example the common ratio was 3:

We can start with any number:

2, 6, 18, 54,162,486, ...

This sequence also has a common ratio of 3, but it starts with 2.

1, 2, 4, 8, 16, 32, 64,128,256, ...

This sequence starts at 1 and has a common ratio of 2.
The pattern is continued by multiplying by 2 each time, like this:

The common ratio can be less than 1:

10, 5, 2.5, 1.25, 0.625, 0.3125, ...

This sequence starts at 10 and has a common ratio of 0.5 (a half).
The pattern is continued by multiplying by 0.5 each time.

But the common ratio can't be 0, as we get a sequence like 1, 0, 0, 0, 0, 0, 0, ...

658,796, 1741, 10006, 10007

There are also many special sequences, here are some of the most common:

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This Triangular Number Sequence is generated from a pattern of dots that form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

Square Numbers

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...

They are the squares of whole numbers:

0 (=0×0) 1 (=1×1) 4 (=2×2) 9 (=3×3) 16 (=4×4)

etc...

Cube Numbers

1, 8, 27, 64,125,216,343,512,729, ...

They are the cubes of the counting numbers (they start at 1):

1 (=1×1×1) 8 (=2×2×2) 27 (=3×3×3) 64 (=4×4×4)

etc...

Fibonacci Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The Fibonacci Sequence is found by adding the two numbers before it together. The 2 is found by adding the two numbers before it (1+1) The 21 is found by adding the two numbers before it (8+13) The next number in the sequence above would be 55 (21+34)

Can you figure out the next few numbers?