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Number sequences & the nth term!
In these types of questions you may be asked to find the next few terms or you may be asked to find the value of a specific term, say the 25th term.
In a sequence of numbers such as this you are given the first three terms. They are:
3, 5, 7 ,_ ,_
You may be asked to fill in the value of the missing blanks (here they are terms 4 & 5). What you should be looking for is a pattern. Here each number is increasing by 2, there is a constant difference of 2:
3, 5, 7, 9, 11…
There are sequences where the difference changes :
2, 3, 5, 8, 12, _, _
Here the difference between each number is increasing by one each time. Can you work out the missing values for the last two terms? (terms 6 & 7)
However there are a few more tricky ones you should be able to identify.
1, 1, 2, 3, 5, 8, 13…… etc
This is called the Fibonacci sequence, generated by adding together the two previous numbers, 1+1=2, 2+1=3, 3+2=5 & so on.
This one is an oscillating sequence:
10, 8, 12, 6, 14, 4, 16, _, _
You can almost treat this as two separate sequences, one being, 10, _, 12, _, 14, _, the other being, 8, _, 6, _, 4, _. Can you figure out the value of the next two terms in the oscillating sequence?
The nth term.
When being asked to find the value of the twenty fifth term, working out each value in turn as we get to it may not be a sensible option & could be very time consuming. What if we are given the first four terms and asked to find the fiftieth?! Clearly it would be easier if we could somehow work out the fiftieth term without having first to work out the 5th, 6th, 7th etc.
We can do this quite easily for some sequences. What we first look for is the difference between the value of each term in the sequence. If it is a constant difference like the example above we make a note of what that difference is. The difference in 3, 5, 7, 9, 11 is 2. We take the differnce and simply write the letter ‘n’ next to it:
Now if we were to try and use this to produce the 1st term we would substitute the letter ‘n’ with the number 1 for the 1st term, (if we wanted the 2nd term we would replace the letter ‘n’ with the number 2 and so on)
Now we have, 2 x 1 which would give us a value of 2. However if we look at the sequence we needed a value of 3, so our expression, 2n, is not finished yet, we need a correction factor. This is simply a number we add or subtract from our expression to give the value we wanted. We can turn our 2 into a 3 simply by adding on 1:
2n + 1
When we try this with the first term we have (2 x 1) + 1 = 3 which is exactly what we needed. If we try it with the 2nd term (2 x 2) + 1 = 5, the 3rd term (2 x 3) +1 = 7 so we know that the nth term is given by:
nth = 2n + 1
To find the value of thr 50th term all we would have to do is replace the n with 50. Why don’t you have a go and see if you can find the value of the 50th term?
A sequence with a constant difference such as this is also known as arithmetic progression.
What about the nth term of a changing difference?
These type of questions are a bit more tricky. Take this sequence for example:
2, 4, 8, 14, 22, 32,…….
Well the difference here is not constant, it’s increasing by 2 each time. We’ll use the formula:
nth = a + (n-1)d + ½ (n-1)(n-2)C
Bit of a monster huh? Still, it’s worth remembering. What do the letters stand for?
a = value of the first term (in our example that would be 2)
d = difference between first two terms (in our example that would be 2 i.e. 4-2=2)
C = the change in the difference (in our example this is also 2)
Then all we need is a bit of substitution & then some simplification.
nth = a + (n-1)d + ½ (n-1)(n-2)C before substitution
nth = 2 + (n-1)2 + ½ (n-1)(n-2)2 after substitution
Now we expand brackets, simplify and end up with:
nth = n2 – n + 2 after simplification
And if we try it for say the 4th term, which we already know, just to see if it works:
4th = 42 – 4 + 2
= 16 – 4 + 2
This is the value of our fourth term. You may want to try it on the other terms to convince yourself then find the value of the 15th term. When you’ve done that why not have a go at this one. Find the value of the 15th term:
1, 4, 9, 16….
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