This document discusses different types of sequences: 1. Arithmetic sequences are sequences where the difference between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common difference. 2. Geometric sequences are sequences where the ratio between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common ratio. 3. Examples are provided to demonstrate how to calculate the nth term and sum of terms for both arithmetic and geometric sequences. Formulas are given for finding individual terms and summing finite and infinite sequences.

1. 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.
Example
an= 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.
Example
Write a sequence whose first term is 2, knowing that each term is the square of the previous term.
2, 4, 16, ...

2. 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 = 8
19 – 11 = 8
27 – 19 = 8
35 – 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) · d
8, 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) · d
a4 = −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.
Example
Calculate 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.
Example
3, 6, 12, 24, 48, ...
6/3 = 2.
12/6 = 2. r = 2.
24/12 = 2.
48/24 = 2.

3. 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
Example
Compute 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:
Example
Calculate the sum of the terms of the sequence:
This sum is related to the periodic numbers.