Arithmetic sequence

Let’s say we have a sequence of numbers 2, 7, 12, 17, 22,…

How can we describe this sequence?

The sequence starts with 2

The difference between a number in the sequence and the previous number in the sequence is always 5

It can be useful to put the sequence in a table

If we label each number in the sequence by n and the value of each number with a_n and the difference between adjacent numbers in the sequence with d then we can put the sequence into a table like this

Now we can write a formula to give the value of any term in the sequence

$a_n = a_1 + (n - 1)d$

Let’s check this works for the 5th term

$a_5 = a_1 + (5 - 1)d$

$a_5 = 2 + 4 \times 5 = 22$

And work out what the 100th term is

$a_{100} = a_1 + (100 - 1)d = 2 + 99 \times 5 = 497$

This type of sequence is called arithmetic because it is related to adding. In the example that we have looked at, we are simply adding 5 to the previous term in the sequence. The word arithmetic comes from a Greek word arithmetike meaning the art of counting.

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Numbers Part Two

Arithmetic sequence