Determine uma progressão aritmética (P.A.) crescente de três termos sabendo que a soma deles é 45 e o produto entre eles é 3240
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Solução!
![P.A:\{(x-r),(x),(x+r)\}\\\\\\
x-r+x+x+r=45\\\\\\\
(x-r).(x).(x+r)=3240](https://tex.z-dn.net/?f=P.A%3A%5C%7B%28x-r%29%2C%28x%29%2C%28x%2Br%29%5C%7D%5C%5C%5C%5C%5C%5C%0Ax-r%2Bx%2Bx%2Br%3D45%5C%5C%5C%5C%5C%5C%5C%0A%28x-r%29.%28x%29.%28x%2Br%29%3D3240 "P.A:\{(x-r),(x),(x+r)\}\\\\\\
x-r+x+x+r=45\\\\\\\
(x-r).(x).(x+r)=3240")
![Soma\\\\\
x-r+x+x+r=45\\\\\
3x=45\\\\\
x= \dfrac{45}{3}\\\\\
\boxed{x=15}](https://tex.z-dn.net/?f=Soma%5C%5C%5C%5C%5C%0Ax-r%2Bx%2Bx%2Br%3D45%5C%5C%5C%5C%5C%0A3x%3D45%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B45%7D%7B3%7D%5C%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D15%7D+ "Soma\\\\\
x-r+x+x+r=45\\\\\
3x=45\\\\\
x= \dfrac{45}{3}\\\\\
\boxed{x=15}")
![Produto!\\\\\\
(x-r).(x).(x+r)=3240\\\\\
( x^{3}+ x^{2} r- x^{2} r-r^{2}x)=3240\\\\\\\
x^{3}-r^{2}x=3240\\\\\
Sendo~~x=15\\\\\\\
(15)^{3} +15r^{2}=3240\\\\\\\\
3375-15r^{2}=3240\\\\\\\\
-15r^{2}=3240-3375\\\\\\\\
15r^{2}=-135\\\\\\\\
r^{2}= \frac{-135}{-15}\\\\\\
r^{2}=9\\\\\
r= \sqrt{9}\\\\\\
r=\pm3\\\\\\\
Para~~que~~a~~P.A~~seja~~crescente~~n\~ao~~podemos~~usar~~-3](https://tex.z-dn.net/?f=Produto%21%5C%5C%5C%5C%5C%5C%0A%28x-r%29.%28x%29.%28x%2Br%29%3D3240%5C%5C%5C%5C%5C%0A%28+x%5E%7B3%7D%2B+x%5E%7B2%7D+r-+x%5E%7B2%7D+r-r%5E%7B2%7Dx%29%3D3240%5C%5C%5C%5C%5C%5C%5C%0A+x%5E%7B3%7D-r%5E%7B2%7Dx%3D3240%5C%5C%5C%5C%5C%0ASendo%7E%7Ex%3D15%5C%5C%5C%5C%5C%5C%5C%0A+%2815%29%5E%7B3%7D+%2B15r%5E%7B2%7D%3D3240%5C%5C%5C%5C%5C%5C%5C%5C%0A+3375-15r%5E%7B2%7D%3D3240%5C%5C%5C%5C%5C%5C%5C%5C%0A-15r%5E%7B2%7D%3D3240-3375%5C%5C%5C%5C%5C%5C%5C%5C%0A15r%5E%7B2%7D%3D-135%5C%5C%5C%5C%5C%5C%5C%5C%0Ar%5E%7B2%7D%3D+%5Cfrac%7B-135%7D%7B-15%7D%5C%5C%5C%5C%5C%5C%0Ar%5E%7B2%7D%3D9%5C%5C%5C%5C%5C%0Ar%3D+%5Csqrt%7B9%7D%5C%5C%5C%5C%5C%5C%0Ar%3D%5Cpm3%5C%5C%5C%5C%5C%5C%5C%0APara%7E%7Eque%7E%7Ea%7E%7EP.A%7E%7Eseja%7E%7Ecrescente%7E%7En%5C%7Eao%7E%7Epodemos%7E%7Eusar%7E%7E-3+++++ "Produto!\\\\\\
(x-r).(x).(x+r)=3240\\\\\
( x^{3}+ x^{2} r- x^{2} r-r^{2}x)=3240\\\\\\\
x^{3}-r^{2}x=3240\\\\\
Sendo~~x=15\\\\\\\
(15)^{3} +15r^{2}=3240\\\\\\\\
3375-15r^{2}=3240\\\\\\\\
-15r^{2}=3240-3375\\\\\\\\
15r^{2}=-135\\\\\\\\
r^{2}= \frac{-135}{-15}\\\\\\
r^{2}=9\\\\\
r= \sqrt{9}\\\\\\
r=\pm3\\\\\\\
Para~~que~~a~~P.A~~seja~~crescente~~n\~ao~~podemos~~usar~~-3")
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