Ensino Médio - Ágebra

por EVARISTO BRITO SANTOS Sex 29 Jun 2018, 18:03

Calcule o valor da expressão:
(2³-1)(3³-1) ... (99³-1)(100³-1)/ (2³+1)(3³+1) ... (99³+1)(100³+1)

por poisedom Sáb 30 Jun 2018, 20:48

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)= \dfrac{3367}{5050} pois

observe que

(a^3-1)=(a+1)(a^2-a+1)
(a^3-1)=(a+1)(a^2-a+a-a+1-1+1)
(a^3-1)=(a+1)[(a^2-2a+1)+(a-1)+1]
(a^3-1)=(a+1)[(a-1)^2+(a-1)+1]

e que
(a^3+1)=(a-1)(a^2+a+1)

e agora note (a-1) é o termo anterior e (a+1) o posterior a a assim temos:

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{7\cdot 26\cdot 63\cdot124 \cdots 941191 \cdot 970298 \cdot 999999}{9\cdot28\cdot65\cdot126\cdots 941193\cdot970300\cdot1000001}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{(1\cdot7)\cdot (2\cdot13)\cdot (3\cdot21)\cdot(4\cdot31) \cdots (97\cdot9703) \cdot (98 \cdot9901) \cdot(99\cdot10101)}{(3\cdot3)\cdot(4\cdot7)\cdot(5\cdot13)\cdot(6\cdot21)\cdots (99\cdot9507)\cdot(100\cdot9703)\cdot(101\cdot9901)}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{(1\cdot 2 \cdot 3 \cdot 4 \cdots 98\cdot99)\cdot (7\cdot13\cdot21\cdot31\cdots 9703\cdot9901\cdot10101)}{(3\cdot 4 \cdot5\cdot6 \cdots99\cdot100\cdot101)\cdot(3\cdot7\cdot13\cdot21\cdots9507\cdot9703\cdot9901)}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{1\cdot2\cdot10101}{100\cdot101\cdot3}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{1\cdot2\cdot3367\cdot 3}{50\cdot 2\cdot101\cdot3}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)=\dfrac{1\cdot3367}{50\cdot101}

\prod_{a = 2}^{100} (a^3-1)/\prod_{a = 2}^{100} (a^3+1)= \dfrac{3367}{5050}

Ensino Médio - Ágebra

Ensino Médio - Ágebra

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