Sequences and progressions

# Sequences

A **sequence** is an ordered set of numbers (or other objects), arranged according to a rule.

For example:

· 1, 2, 3, 4, 5, …

· 2, 4, 6, 8, 10, …

· 1, 4, 9, 16, 25, …

· 32, 16, 8, 4, 2, …

· 1, 1, 2, 3, 5, 8, 13, …

Each number of the sequence is called

**term**, and we represent it as a_{i}. We call**general term**of a sequence, an, the expression which represents any term of the sequence.For example:

· 1, 2, 3, 4, 5, … a

_{n}= n, then a_{100}= 100· 2, 4, 6, 8, 10, … a

_{n}= 2n, then a_{25}= 50· 1, 4, 9, 16, 25, … a

_{n}= n^{2}, then a_{12}= 144· 32, 16, 8, 4, 2, … a

_{n}= 2^{6-n}, then a_{10}= 2^{-4}= 1/16Some sequences are

**recurring**, because we obtain each term from the previous one. For example: 1, 1, 2, 3, 5, 8, 13, … a

_{n}= a_{n-1}+ a_{n-2 }a_{1}= a_{2}= 1This is the** Fibonacci sequence**.

**Exercises**:

1.- Find out the general term of these sequences:

a) 3, 6, 9, 12, 15, .....

2.- Write the first five terms of the sequences whose general terms are:

a) a_{n} = n^{2} + 1

b) a_{n} = 3 - n

Solutions: 1.- a) a_{n} = 3n; b) a_{n} = 1/n; c) a_{n} = n/(n+1)

2.- a) 2, 5, 10, 17, 26,... b) 2, 1, 0, -1, -2,... c) 2, 3/4, 4/9, 5/16, 6/25,...

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