Mathematics

Sequences

Concept of a sequence and general term. Study of Arithmetic and Geometric Progressions, including finite and infinite sum formulas. Special sequences: Fibonacci and Harmonic.

A sequence is an ordered, infinite list of numbers that follow a rule or pattern. The terms are denoted as {a1,a2,a3,…,an}\{a_1, a_2, a_3, \dots, a_n\}, where the subscript indicates its position in the list. Examples:
{an}={1,3,5,7,… }{bn}={1,2,4,8,16,… }\begin{aligned} \{a_n\} &= \{1, 3, 5, 7, \dots\} \\ \{b_n\} &= \{1, 2, 4, 8, 16, \dots\} \end{aligned}
The general term (or nn-th term), ana_n, of a sequence is the rule or mathematical expression that allows us to calculate the value of any term given its position. Examples:
{1,3,5,7,… }⇒an=2n−1{−1,2,−3,4,… }⇒bn=n(−1)n{3,3,3,3,… }⇒cn=3\begin{matrix} \{1, 3, 5, 7, \dots\} & \Rightarrow & a_n = 2n-1 \\ \{-1, 2, -3, 4, \dots\} & \Rightarrow & b_n = n(-1)^n \\ \{3, 3, 3, 3, \dots\} & \Rightarrow & c_n = 3 \end{matrix}
An arithmetic sequence is a sequence in which, starting from an initial term a1a_1, each term is obtained by adding a constant number dd (called the common difference) to the previous term. Example:
{1,3,5,7,… }witha1=1  ;d=2\{1, 3, 5, 7, \dots\} \qquad \text{with} \qquad a_1 = 1 \;; \quad d = 2
The general term of an arithmetic sequence is given by:
an=a1+(n−1)da_n = a_1 + (n-1)d
The sum of the first nn terms is given by:
Sn=n(a1+an)2S_n = \dfrac{n(a_1 + a_n)}{2}
A geometric sequence is a sequence in which, starting from an initial term a1a_1, each term is obtained by multiplying the previous one by a constant number rr, called the common ratio. Ex.:
{1,2,4,8,… }witha1=1  ;r=2\{1, 2, 4, 8, \dots\} \qquad \text{with} \qquad a_1 = 1 \;; \quad r = 2
The general term is given by:
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
The sum of the first nn terms is given by:
Sn=a1(rn−1)r−1withr≠1S_n = \dfrac{a_1(r^n - 1)}{r - 1} \qquad \text{with} \qquad r \neq 1
The product of the first nn terms is given by:
Pn=±(a1⋅an)n=±a1n⋅rn(n−1)2P_n = \pm\sqrt{(a_1 \cdot a_n)^n} = \pm a_1^n \cdot r^{\frac{n(n-1)}{2}}
If −1<r<1-1 < r < 1, the geometric sequence is convergent and we can calculate the sum of its infinite terms:
S∞=a11−rS_\infty = \dfrac{a_1}{1 - r}
Fibonacci Sequence: starting with a1=1a_1 = 1 and a2=1a_2 = 1, each subsequent term is the sum of the two previous terms:
{1,1,2,3,5,8,13,… }\{1, 1, 2, 3, 5, 8, 13, \dots\}
The general term is recursive, meaning it requires knowing the previous terms to find the next one:
an=an−1+an−2a_n = a_{n-1} + a_{n-2}
There is an explicit expression that allows us to find any term directly, called Binet's Formula:
an=15[(1+52)n−(1−52)n]a_n = \dfrac{1}{\sqrt{5}} \left[ \left( \dfrac{1+\sqrt{5}}{2} \right)^n - \left( \dfrac{1-\sqrt{5}}{2} \right)^n \right]
Fibonacci Golden Spiral
A harmonic sequence is one in which each term is the multiplicative inverse (reciprocal) of an arithmetic sequence that does not contain zero. For example, given the arithmetic sequence:
{1,3,5,7,… }withan=2n−1\{1, 3, 5, 7, \dots\} \qquad \text{with} \qquad a_n = 2n-1
The corresponding harmonic sequence is:
{1,13,15,17,… }withan=12n−1\left\{ 1, \dfrac{1}{3}, \dfrac{1}{5}, \dfrac{1}{7}, \dots \right\} \qquad \text{with} \qquad a_n = \dfrac{1}{2n-1}
The standard Harmonic Sequence:
{1,12,13,14,… }withan=1n\left\{ 1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dots \right\} \qquad \text{with} \qquad a_n = \dfrac{1}{n}
It represents the harmonics of sound waves produced by musical instruments. For example, when a guitar string is plucked, the entire string vibrates from end to end (producing the fundamental frequency). At the same time, it also vibrates in exact halves, thirds, fourths... The wavelengths of these secondary waves formed on the string follow this exact mathematical sequence. These higher-pitched secondary sounds that accompany the fundamental note are called Harmonics (Overtones).