You probably saw questions ‘What is the next number in the sequence?’ quite early in studying numbers.
Next number in a sequence example
2 4 6 8 10 .. what is the next number?
Let’s just ignore the people who can see the answer straight off, and look for a method.
Look for the GAPS between each number. That is always the best start. In this example, the gap between each number is 2 – To put it another way, the numbers are going UP by 2 each time.
Once we have spotted that, we can move on through the sequence, adding two onto the last number each time.
… 12 14 16
Where we can get clever though is trying to find a general term in this sequence. This is a step up in effort, for sure.
What we do here is give all the numbers in a sequence a ‘place in the sequence’ which we do, in true algebra fashion, by using a letter. Its normal to use n in sequences. Questions will usually ask ‘find the nth’ term?’
We say we the first term in the sequence is n=1, then n=2 for the second, and so on. The general term then uses n in a formula. Let’s see how its done
Finding the nth term
The first thing is to see the gap, as we saw before. In the first example, the gap was 2
1 4 7 10 13
In this sequence the gap is 3
So we start our nth term formula by putting this gap number in front of the n. 2n for the first example, 3n for this example.
So – does ‘3n’ give the sequence we’ve been asked to investigate?
No, because that sequence is
3 6 9 12 15 ..
But we can compare the two sequences – something we do a lot when looking for nth term formulas – and see our sequence is 2 less for eeach term than 3n: 3 6 9 …
so we have our formula 3n – 2
We can check, say, the 5th term – 5 x 3 – 2 = 15 – 2 = 13. That matches what we were given. We can now confidently predict the 100th term
100 x 3 – 2 = 200 – 2 = 298
Is the number in the sequence?
Another common question is – is 100 in the sequence 1 4 7 10….
This question has not asked you to find the nth term – but that is the route to finding the answer.
We have already found the nth term for this sequence. This means we need to find n where
3n – 2 = 100.
We start to solve this like an equation, by taking 2 from both sides
3n = 98.
Now we hit a hitch. n is not going to be a whole number because 3 is not a factor of 98. In sequences we are only interested in cases where n IS a whole number.
from this we can say that 100 is NOT in the sequence because there is no n where 3n – 2 = 100
For more practice, see this website