Difference between revisions of "Somatório e Produtório"

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Para <math>k+1</math>, assumindo o lado esquerdo da equação, temos:
 
Para <math>k+1</math>, assumindo o lado esquerdo da equação, temos:
  
−
<math> \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot f(k+1) + \sum_{n=s}^k C\cdot F(n)</math>, pela definição de somatório.
+
<math> \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot f(k+1) + \sum_{n=s}^k C\cdot f(n)</math>, pela definição de somatório.
  
  
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Portanto:
 
Portanto:
  
−
<math> \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) </math>, onde C é uma constante, <math>\forall t \in N</math>.
+
<math> \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) </math>, onde C é uma constante, <math>\forall s, t \in N</math>.
  
 +
 +
 +
=== Mudança de índices ===
 +
<math> \sum_{n=s}^t f(n) = \sum_{n=s+1}^{t+1} f(n-1) </math>
 +
 +
===== Passo base: s = t =====
 +
<math> \sum_{n=s}^t f(n) = f(n) = \sum_{n=s+1}^{t+1} f(n-1) </math>, pela definição de somatório.
 +
 +
===== Passo indutivo: s < t =====
 +
 +
Suponha que para um <math>k \in N, k > s</math> arbitrário:
 +
 +
<math> \sum_{n=s}^k f(n) = \sum_{n=s+1}^{k+1} f(n-1) </math> (Hipótese de indução)
 +
 +
 +
Para <math>k+1</math>, assumindo o lado esquerdo da equação, temos:
 +
 +
<math> \sum_{n=s}^{k+1} f(n) = f(k+1) + \sum_{n=s}^k f(n)</math>, pela definição de somatório.
 +
 +
 +
Aplicando a HI:
 +
 +
<math> \sum_{n=s}^{k+1} f(n) = f(k+1) + \sum_{n=s+1}^{k+1} f(n-1)</math>
 +
 +
 +
Expandindo <math>k-s</math> vezes:
 +
 +
<math> \sum_{n=s}^{k+1} f(n) = f(k+1) + f(k+1-1) + f(k-1) + ... + f(s-1) + f(s+1-1)</math>
 +
 +
<math> \sum_{n=s}^{k+1} f(n) = f(k+1) + f(k) + f(k-1) + ... + f(s-1) + f(s)</math>
 +
 +
<math> \sum_{n=s}^{k+1} f(n) = \sum_{n=s+1}^{k+2} f(n-1)</math>, uma vez que existem <math>k+2</math> termos.
 +
 +
 +
Portanto:
 +
 +
<math> \sum_{n=s}^t f(n) = \sum_{n=s+1}^{t+1} f(n-1) \forall s, t \in N</math>.
 
----
 
----
  

Revision as of 02:00, 9 December 2015

Propriedades de Somatório

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) } , onde C é uma constante.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right] }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right] }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^n_{i = m} f(i) = \sum^{n+p}_{i = m+p} f(i-p) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{n=s}^{t} j = \sum\limits_{n=1}^{t} j - \sum\limits_{n=1}^{s-1} j }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n) = \sum_{n=s}^t f(n)} , note que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \leq j \leq t }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=m}^n i = \frac{n(n+1)}{2} - \frac{m(m-1)}{2} = \frac{(n+1-m)(n+m)}{2},} progressão aritmética.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{k=0}^{n-1}{2^k} = 2^n-1 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=s}^m\sum_{j=t}^n {a_i}{c_j} = \sum_{i=s}^m a_i \cdot \sum_{j=t}^n c_j }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^n i^3 = \left(\sum_{i=0}^n i\right)^2 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=m}^{n-1} a^i = \frac{a^m-a^n}{1-a} (m < n) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=0}^{n-1} a^i = \frac{1-a^n}{1-a} }


Principais representações

Soma simples

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{n} x_i = x_1+x_2+...+x_n}

Soma de quadrados

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{n} x_i^2 = x_1^2+x_2^2+...+x_n^2}

Quadrado da soma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\sum_{i=1}^{n} x_i)^2 = (x_1+x_2+...+x_n)^2}

Soma de produtos

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^{n} x_iy_i = x_1y_1+x_2y_2+...+x_ny_n}

Produtos das somas

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\sum_{i=1}^{n} x_i)(\sum_{j=1}^{m} y_j) = (x_1+x_2+...+x_n)(y_1+y_2+...+y_n)}

Aplicação das Propriedades

Alguns exemplos de aplicações das propriedades do somatório:


Provas de algumas propriedades

Multiplicação por constante

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) } , onde C é uma constante.

Passo base: s = t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t C\cdot f(n) = C\cdot f(n) } , pela definição de somatório.

Passo indutivo: s < t

Suponha que para um Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \in N, k > s} arbitrário:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^k C\cdot f(n) = C\cdot \sum_{n=s}^k f(n) } (Hipótese de indução)


Para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+1} , assumindo o lado esquerdo da equação, temos:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot f(k+1) + \sum_{n=s}^k C\cdot f(n)} , pela definição de somatório.


Aplicando a HI:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot f(k+1) + C\cdot \sum_{n=s}^k f(n)}


Expandindo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k-s} vezes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot (f(k+1)) + C\cdot (f(k) + f(k-1) + ... + f(s+1) + f(s))}


Colocando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} em evidência:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot (f(k+1) + f(k) + f(k-1) + ... + f(s+1) + f(s))}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} C\cdot f(n) = C\cdot \sum_{n=s}^{k+1} f(n) }


Portanto:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) } , onde C é uma constante, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall s, t \in N} .


Mudança de índices

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t f(n) = \sum_{n=s+1}^{t+1} f(n-1) }

Passo base: s = t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t f(n) = f(n) = \sum_{n=s+1}^{t+1} f(n-1) } , pela definição de somatório.

Passo indutivo: s < t

Suponha que para um Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \in N, k > s} arbitrário:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^k f(n) = \sum_{n=s+1}^{k+1} f(n-1) } (Hipótese de indução)


Para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+1} , assumindo o lado esquerdo da equação, temos:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} f(n) = f(k+1) + \sum_{n=s}^k f(n)} , pela definição de somatório.


Aplicando a HI:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} f(n) = f(k+1) + \sum_{n=s+1}^{k+1} f(n-1)}


Expandindo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k-s} vezes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} f(n) = f(k+1) + f(k+1-1) + f(k-1) + ... + f(s-1) + f(s+1-1)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} f(n) = f(k+1) + f(k) + f(k-1) + ... + f(s-1) + f(s)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^{k+1} f(n) = \sum_{n=s+1}^{k+2} f(n-1)} , uma vez que existem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+2} termos.


Portanto:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=s}^t f(n) = \sum_{n=s+1}^{t+1} f(n-1) \forall s, t \in N} .

Somatório em Linguagem Funcional

Elixir[1]

defmodule FMC do
  def somatorio(start \\0, finish, callback)

  def somatorio(start, finish, callback) when start == finish do
    callback.(start)
  end

  def somatorio(start, finish, callback) do
    _somatorio(Enum.to_list(start..finish), callback)
  end

  defp _somatorio([], _), do: 0
  defp _somatorio([head | tail], callback) do
    callback.(head) + _somatorio(tail, callback)
  end
end

Referências

  1. ↑ https://github.com/jaimerson/fmc-elixir-somatorio
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