math
10th Grade - Sequences

10th Grade lesson

5 - Sequences


Sequences

A 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 1st term of the sequence its first term, 2nd term of the sequence its second term, 3rd term of the sequence 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 sequence formula.
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 :
sequence bye recursion

A sequence is increasing if sequence is always greater than sequence. To prove that a sequence is increasing, you can calculate sequence and prove that the result is positive. For the first example, calcul variation sequence hence the sequence is increasing.

A sequence is bounded above if there is a number M such as for every number n, bounded sequence (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 sequences

A sequence is arithmetic when aritmetic sequence-aritmetic sequence is constant. The common difference of an arithmetic sequence is the number r such that proprerty aritmetic sequence. 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 :
Hence :

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.
calcul termes aritmetic sequence (premier terme de la sequence is the 1st term, then sequence is the 47th term !)

formule aritmetic sequence

By adding the two equalities then dividing the result by two we obtain :

somme des termes aritmetic sequence

In general :
aritmetic sequence


Geometric sequences

Nothing to do with geometry. A sequence is geometric when the quotient geometric sequence is constant.
Its value q verifies ratio geometric sequence. q is also called the common ratio of the geometric sequence. The second sequence of this page is geometric, its first term is first term and its common ratio is (-2). To calculate  sequence geometric, we notice that
sequence geometric

In general,
formule termes sequence geometric
Thanks to this formula, sequence calculation geometric.

There's also a formula that enables us to work out the sum of the terms of a geometric sequence :

geometric sequence sum formula

We wan calculate the sum of the first 4 terms of the geometric sequence of the following example :
exemple calcul de la somme des termes sequence geometric

which is equal to 5 - 10 + 20 - 40 = - 25.



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Sequences

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