I.E.S. MARÍA BELLIDO - BAILÉN

BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA

SEQUENCES

1. SEQUENCES

A sequence is a set of numbers arranged one after another.

3, 6, 9,..., 3n

The numbers a1, a2, a3 ,..., an are called terms or elements of the sequence.The subscript is the set of positive integers 1, 2, 3, ... The subscript indicates the place that a term occupies in the sequence. The nth term is denoted by an.

2. CALCULATION OF A SEQUENCE

By the Nth Term

an is a criterion that allows us to calculate any term of the sequence.

Examplean= 2n − 1 a1 = 2 ·1 − 1 = 1 a2 = 2 ·2 − 1 = 3 a3= 2 ·3 − 1 = 5 a4 = 2 ·4 − 1 = 7 1, 3, 5, 7, ..., 2n −1

Not all sequences have a general term. For example, the sequence of prime numbers:2, 3, 5, 7, 11, 13, 17, 19, 23,...

By a Recursive Formula

A term is obtained by operating with the previous terms.

ExampleWrite a sequence whose first term is 2, knowing that each term is the square of the previous term.2, 4, 16, ...

3. ARITHMETIC SEQUENCE

An arithmetic sequence or arithmetical progression is a sequence of numbers such that the differences between successive terms is a constant, d, called the common difference. In other words, each term is obtained by adding d of previous.

an - an-1 = d

EXAMPLE: Find the common difference and the next term of the following sequence:3, 11, 19, 27, 35,...

To find the common difference, I have to subtract a pair of terms. It doesn't matter which pair I pick, as long as they're right next to each other:

11 – 3 = 819 – 11 = 827 – 19 = 835 – 27 = 8

The difference is always 8, so d = 8. Then the next term is 35 + 8 = 43.

Nth Term of an Arithmetic Sequence

Example 1:If the 1st term is known.an = a1 + (n − 1) · d8, 3, −2, −7, −12, ..an= 8 + (n−1) (−5) = 8 −5n +5 = = −5n + 13

Example 2:If the value that occupies any other term of the sequence is known.an = ak + (n − k) · da4 = −7 and d = −5 an = −7+ (n − 4) · (−5)= −7 −5n +20 = −5n + 13

Sum of an Arithmetic Sequenceries

An arithmetic series is the sum of an arithmetic sequence.

ExampleCalculate the sum of the first 5 terms of the sequence: 8, 3, −2, −7, −12, ...

4. GEOMETRIC SEQUENCE

A geometric sequence or geometric progression is a sequence of numbers such that the ratios between successive terms is a constant r, called the common ratio. In other words, each term is obtained from the previous term multiplying above the amount r.

Example3, 6, 12, 24, 48, ...6/3 = 2.12/6 = 2. r = 2. 24/12 = 2. 48/24 = 2.

Nth Term of a Geometric Sequence

Example 1:If the 1st term is known.an = a1 · rn-1

3, 6, 12, 24, 48, ..an = 3 · 2n-1 = 3 · 2n · 2-1 = (3/2)· 2n

Example 2:If the value that occupies any other term of the sequence is known.an = ak · rn-k

a4= 24, k=4 and r=2.an = a4 · rn-4

an = 24· 2n-4= (24/16)· 2n = (3/2) · 2n

Sum of a geometric sequence

ExampleCompute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, …

Sum of Infinite terms in a Geometric sequence

If −1 < r < 1 the infinite geometric series converges to a specific value:

ExampleCalculate the sum of the terms of the sequence:

This sum is related to the periodic numbers.