Difference between revisions of "Somatório e Produtório"

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Alguns exemplos de aplicações das propriedades do somatório:
 
Alguns exemplos de aplicações das propriedades do somatório:
  
 +
=== Exemplo 1 ===
  
 +
Utilize  as propriedades de notação  de  somatório e,
 +
possivelmente, mudança  de índice  para deduzir que
 +
<math>\sum_{j=1}^n (a_j - a_{j-1})</math> é igual a <math>a_n - a_0</math>,
 +
onde <math>(a_i )_{i=0}^{\infty}</math> é  uma sequência  de números  reais.
 +
Este  tipo de  soma  é bastante conhecida em Matemática como ''soma telescópica''.
 +
 +
==== Resolução ====
 +
 +
<math> \sum_{j=1}^n (a_j - a_{j-1}) = (a_n - a_{n-1}) + \sum_{j=1}^{n-1}</math>
 +
 +
 +
Expandindo <math>n</math> vezes:
 +
 +
<math> \sum_{j=1}^n (a_j - a_{j-1}) = (a_n - a_{n-1}) + (a_{n-1} - a_{n-2}) + ... + (a_2 - a_1) + (a_1 - a_0)</math>
 +
 +
<math> \sum_{j=1}^n (a_j - a_{j-1}) = a_n - \cancel{a_{n-1}} + \cancel{a_{n-1}} - \cancel{a_{n-2}} + ... + \cancel{a_2} - \cancel{a_1} + \cancel{a_1} - a_0</math>
 +
 +
<math> \sum_{j=1}^n (a_j - a_{j-1}) = a_n - a_0 </math>
  
 
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Revision as of 14:34, 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:

Exemplo 1

Utilize as propriedades de notação de somatório e, possivelmente, mudança de índice para deduzir 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 \sum_{j=1}^n (a_j - a_{j-1})} é igual a 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 a_n - a_0} , onde 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 (a_i )_{i=0}^{\infty}} é uma sequência de números reais. Este tipo de soma é bastante conhecida em Matemática como soma telescópica.

Resolução

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_{j=1}^n (a_j - a_{j-1}) = (a_n - a_{n-1}) + \sum_{j=1}^{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 n} 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_{j=1}^n (a_j - a_{j-1}) = (a_n - a_{n-1}) + (a_{n-1} - a_{n-2}) + ... + (a_2 - a_1) + (a_1 - a_0)}

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_{j=1}^n (a_j - a_{j-1}) = a_n - \cancel{a_{n-1}} + \cancel{a_{n-1}} - \cancel{a_{n-2}} + ... + \cancel{a_2} - \cancel{a_1} + \cancel{a_1} - a_0}

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_{j=1}^n (a_j - a_{j-1}) = a_n - a_0 }

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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