There are many many patterns in the world around us; from a leopard’s spots to the time when the sun rises. Mathematics allows us to describe perfectly these patterns and many more
A sequence is a certain pattern of numbers; and your job is to look for the pattern between all of these numbers. For example let’s look at the pattern of these numbers:
$$ 5,9,13,17, ... $$
We can see here that the pattern starts with the number 5 and then add 4. Although we used words to describe this pattern, we can also use algebra:
$$ 4n + 1 $$
We use n here to mean the position number, which correpsonds to the number in the sequence like this:
[Md table]
We can work out the nth term of a sequence (as this is known) by using the following formula:
$$ a + d(n - 1) $$
a is the first term number, d is the common difference of the pattern (+4 is the common difference here). Let’s use the example to find the nth term for another sequence:
$$ 3, 5, 7, 9, ... $$
$$ \color{red}{3 + 2(n - 1)} = 3 + 2n - 2 = 2n + 1 $$
There are a lot of cool things we can do once we have the nth term of a sequence. For example the nth term is like a list of multiples and we can tell if a number is part of the sequence by setting the nth term equal to the number and finding the value of n, if n is an integer it is part of the sequence. Let’s see if 23 is in the sequence above:
$$ 2n + 1 \color{red}{-1} = 23 \color{red}{-1} $$ $$ \dfrac{2n}{\color{red}{2}} = \dfrac{22}{\color{red}{2}} = 11 $$
So we can tell that 23 is in the sequence (with a position of 11). Sequences continue on forever, however we can find what a certain number of the terms add up to using this formula:
$$ \tfrac{n}{2}[2a + d(n-1)] $$
Let’s use this formula to find the sum of the first 4 terms of the series we just looked at:
$$ \tfrac{\color{red}{4}}{2}[2(\color{red}{3}) + \color{red}{2}(\color{red}{4}-1)] = 2 \times \color{red}{12} = 24 = 3 + 5 + 7 + 9 $$
More coming soon