Write the nth term of a sequence or a series of diagrams
A number sequence is a list of numbers, related by some pattern.
Sometimes we want to know the n^{th }term (which is just the n^{th} number) in the sequence, without writing out the whole sequence, which can be time consuming.
Let's look at a real life example.
Imagine a shop that has just opened.
Say in the first week it took in £1000, then in the second week it took in £1200, then in the third £1400, and the fifth £1600.
You can see that there’s a pattern developing, every week the takings increase by £200.
The owner wants to predict what will happen in the future.
There are a few mathematical tools we can use to help us.
First we set out the information in a way that’s clear to read:
Week 1 2 3 4
Money(£) 1000 1200 1400 1600
To turn it into ‘maths speak’ we’ll change the labels
N 1 2 3 4
Nth Term 1000 1200 1400 1600
Now say I want to predict how much money we’ll take next week. Well, we look at how much it’s going up by each time (the difference in each number). You can see that it goes up by 200 each time. We call this the first order difference. The nth term rule helps us with this problem!
So in week 5 (when N is 5), we’ll expect to take in £1600 + £200 = £1800. Easy. But what about in week 100? Or what about in 5 years time, looking at week 250? We’d be here for a while adding £200 again and again, so there’s a simpler way. We can use algebra to work out a rule for the pattern, and it’s pretty straight forward!
Step 1: Take the first order difference (200) and write n after it: 200n
Step 2: Then ask what you need to add to or subtract from 200 to turn it into our first term (1000). We need to add 800, so our expression becomes:
200n + 800
Now we can use that work out the 100th term by substituting n=100
200 x 100 + 800 = 20,000 + 800 = 20,800
For a real world situation like this one, we turn our mathsy ‘Nth term rule’ into a formula, using our original words or language, e.g.
T = 200w + 800
Where T is takings (in £) and w is the week number since opening.
That said, when looking at real world situations, patterns are never this straight forward, and you should never expect to simply keep taking in an extra £200 each week, the number sequencing and series formula will be more intricate!
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