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Sequences Overview
notation | square
numbers | cubic numbers
Notation
In mathematics, a sequence is a set of sequentially ordered numbers or
objects. They are often related by a rule from which a particular term
might be calculated from previous terms, or from its position in the sequence.
The simplest sequence is the natural numbers, where each number is the
same as its position in the sequence. We can write that as a "rule"
by assigning "n" to its position. Then the "n"th number
will be xn.
1, 2, 3, 4, 5, 6, ...
xn = n
We can also write a rule for calculating the value of each number from
the value of the immediately preceeding number (the one just before it).
In the case of natural numbers:
xn+1 = xn + 1
Square numbers
Square numbers are numbers which are the result of a smaller
whole number multiplied by itself. 9 is a square number made by
multiplying 3 by itself: 3 x 3 = 9. If you have a square number
of small square tiles you can arrange them into a larger square.
Let's look at the sequence of square numbers and its rules.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
xn = n2
xn+1 = ... 
Trying to find a rule for finding xn+1 the first
thing we notice is that the differences between consecutive numbers
itself forms a simple sequence.
1, 3, 5, 7, 9, 11, ...
yn = 2n – 1
yn+1 = yn +
2
∴ yn+1 = (2n – 1) + 2
yn+1 = 2n + 1
So now we know the difference between consecutive numbers, and
we can put that into the equation for our square numbers sequence.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
xn = n2
xn+1 = xn + 2n + 1
∴ xn+1 = n2 + 2n + 1
Square trivia:
- Squares of even numbers are even, for example 6 x 6 = 36.
- Squares of odd numbers are odd. For example, 7 x 7 = 49.
- A square number can only end with digits 00, 1, 4, 6, 9, or
25.
- The square of n is the sum of all odd numbers less than 2n.
- The sum of any two consecutive triangular numbers is a square
number. (Triangular numbers – a very interesting sequence – can be arranged in a triangle: 1, 3,
6, 10, 15, 21, ...)
- When an object falls (ignoring air resistance) the distance
it travels in each fixed amount of time follows the square numbers.
See the picture at right.
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Cubic numbers
Cubic numbers are those which result from multiplying a whole number
by itself then by itself again. For example, 8 is a cubic number made
from 2 x 2 x 2. If you have a cubic number of small cubes you can stack
them together to make a larger cube.
1, 8, 27, 64, 125, 216, ...
xn = n3
For example, if a Rubik's Cube was made of individual small cubes stacked together, there would be 27 small cubes. 3 x 3 x 3 = 27.
The 6th cubic number, 216, is the reason why small
spherical magnets are sold in quantities of 216 – so they
can be grouped into a 6 x 6 x 6 cube. (Small spherical magnets are now illegal to buy in or import into New Zealand.)
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