SequencesA sequence is a sequence of numbers that follow each other in a logical order.
1,3,5,7,9,11,13,15,17,19, etc.... and
5, -10, 20, -40, 80, -160, etc.... are sequences
Call u the sequence. We denote by its first term, its second term, its third term, etc. There are two ways to write a sequence without writing all the terms :
Let's give the formula for the first example, it is .
You can also write it by giving its first term and the relationship between two terms that follow each other. It is the recursive definition. For the second sequence, we obtain :
A sequence is increasing if is always greater than . To prove that a sequence is increasing, you can calculate and prove that the result is positive. For the first example, hence the sequence is increasing.
A sequence is bounded above if there is a number M such as for every number n, (all the terms are smaller than M). M is called a majorant of the sequence. You can define the same way a bounded below sequence, which is a sequence with a minorant. A bounded sequence is a sequence that is bounded above and below. The two sequences in the example are not bounded.
Arithmetic sequencesA sequence is arithmetic when - is constant. The common difference of an arithmetic sequence is the number r such that . The first example is an arithmetic sequence whose first term is 1 and common difference is 2 (we have to give the first term if we want to know the sequence).
To calculate , we notice that :
Thanks to the formula we got .
There is a technique that enables us to calculate the sum of the 47 first terms of an arithmetic sequence.
( is the 1st term, then is the 47th term !)
By adding the two equalities then dividing the result by two we obtain :
In general :
Geometric sequencesNothing to do with geometry. A sequence is geometric when the quotient is constant.
Its value q verifies . q is also called the common ratio of the geometric sequence. The second sequence of this page is geometric, its first term is and its common ratio is (-2). To calculate , we notice that
Thanks to this formula, .
There's also a formula that enables us to work out the sum of the terms of a geometric sequence :
We wan calculate the sum of the first 4 terms of the geometric sequence of the following example :
which is equal to 5 - 10 + 20 - 40 = - 25.