Arithmetic sequences

An arithmetic sequence is a sequence of numbers all separated by same amount. So, the difference between every number in the sequence is the same. An example of this would be the number series: 

{1, 5, 9, 13, 17}

You can see that the difference between every number in this list is 4, as the first term is 1 and if you add 4, you get 5 which is the second term.

It can also go the other way, so that instead of adding you subtract. This would mean that the difference is negative. Using the same example, we would get:

{17, 13, 9, 5, 1}

It is the same difference, 4, but negative, so -4.

If you are presented with the sequence:

{1, 5, 9, 13, 17}

and if asked to find the 13th term, or even the 100th term, you could do it by hand, adding find the next number over and over again, or you could use a formula. In general, if presented with an arithmetic sequence, we have:

un = u1 + (n-1)d

where un is the term you want to find, u1 is the first term in the sequence, n is the term number and d is difference. Continuing with the example, to find the 13th term you do:

u13 = u1 + (13-1) × d

u13 = 1 + (12) × 4

u13 = 1 + 48

u13 = 49

Often you might not be given all the information needed to find the answer. If that's the case, you need to use the general formula to find what you need. For example, given u1 = 16, d = 3, and the nth term is 73, what is n?

Here you have the first term, the difference and the nth term, so all you need is to find the what term 73 is. If we substitute what we have in with the formula, we get:

  73 = 16 + (n-1) × 3

which is a quite simple linear equation, and by some simple rearrangement we have:

3n = 60

divide by 3 on both sides

n = 20

73 is the 20th term in the sequence.

The same can be done for the other variables, although there is a difference trick for finding the difference if not given. If you have the value of two terms, you can find the difference between them, and divide the answer by the difference between the terms numbers you get the difference in the sequence. Example, given the first term of sequence is 1 and the fifth is 17, then the difference is 

d = (17 - 1)/(5 - 1)

d = 16/4 = 4

The sequence from first to fifth term would therefore be:

{1, 5, 9, 13, 17}

We want to find the 18th term thus we know that n is 18. We were also given the values of the first term and the difference in the sequence. We therefore have:

u18 = 13 + (18-1) × 5.5

u18 = 13 + 17 × 5.5

u18 = 13 + 93.5

u18 = 106.5

The 18th term in the sequence is 106.5