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The next number in a sequence

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

More Practice

For more practice, see this website

 

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