Toán Lớp 7: cho 4 số a1,a2,a3,a4 khác 0 thỏa mãn a2^2=a1*a3 và a3^2=a2*a4.chứng minh a1^3+a2^3+a3^3/a2^3+a3^3+a4^3=a1/a4
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Toán Lớp 7: cho 4 số a1,a2,a3,a4 khác 0 thỏa mãn a2^2=a1*a3 và a3^2=a2*a4.chứng minh a1^3+a2^3+a3^3/a2^3+a3^3+a4^3=a1/a4
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a_2^2 = {a_1}.{a_3} \Leftrightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{a_2}}}{{{a_3}}}\\
a_3^2 = {a_2}.{a_4} \Leftrightarrow \dfrac{{{a_2}}}{{{a_3}}} = \dfrac{{{a_3}}}{{{a_4}}}\\
\Leftrightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{a_2}}}{{{a_3}}} = \dfrac{{{a_3}}}{{{a_4}}} = k\left( {k \ne 0} \right)\\
\Leftrightarrow \left\{ \begin{array}{l}
{a_1} = {a_2}.k\\
{a_2} = {a_3}.k\\
{a_3} = {a_4}.k
\end{array} \right.\\
\Leftrightarrow {a_1} = {a_3}.k.k = {a_4}.k.{k^2} = {a_4}.{k^3}\\
\Leftrightarrow \dfrac{{{a_1}}}{{{a_4}}} = {k^3}\\
+ \dfrac{{a_1^3 + a_2^3 + a_3^3}}{{a_2^3 + a_3^3 + a_4^4}} = \dfrac{{a_2^3.{k^3} + a_3^3.{k^3} + a_4^3.{k^3}}}{{a_2^3 + a_3^3 + a_4^4}}\\
= \dfrac{{\left( {a_2^3 + a_3^3 + a_4^4} \right).{k^3}}}{{a_2^3 + a_3^3 + a_4^4}} = {k^3}\\
\Leftrightarrow \dfrac{{a_1^3 + a_2^3 + a_3^3}}{{a_2^3 + a_3^3 + a_4^4}} = \dfrac{{{a_1}}}{{{a_4}}}
\end{array}$