An arithmetic sequence is a sequence such that the difference between successive terms is constant.
2, 6, 10, 14, 18
has a constant common difference term 4. We add 4 to each term to get the next term. We can write down the rule:
Arithmetic sequences can be defined iteratively, so that each term is calculated from the last term, or in closed form, such that we have a formula for the n th term.
For the above sequence the closed form would beIn general the closed form for the nth term can be found from the expressionwhere is the first term andis the common difference. There is also a formula for the sum of the first n terms:The fomulae may be used in the following ways:
A sequence starts 5, 9, 13, 17, 21. Find the sum of all the terms between 100 and 200.
the first termis 5 and the common differenceis 4. We need to find how many terms are less than 100 and how many are less than 200.
terms are less than or equal to 100.
terms are less than or equal to 200.
The difference ofandwill give the sum of all the terms between 100 and 200, so the answer is 4949-1224=3725
Example: for the sequence 5, 12, 19, 26, 33 find the sum of all the terms less than 200.
The first termis 5 and the common differenceis 7.