1.
[tex]P.A(m, 1, n)[/tex]
[tex]r = a_{2} - a_{1}[/tex]
[tex]r = a_{3} - a_{2}[/tex]
[tex]a_{2} - a_{1} = a_{3} - a_{2}[/tex]
[tex]a_{2} + a_{2} = a_{3} + a_{1}[/tex]
[tex]2*a_{2} = a_{1} + a_{3}[/tex]
[tex]2*1 = m + n[/tex]
[tex]m + n = 2[/tex]
[tex]m = 2 - n[/tex]
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[tex]PG(m, n, -8)[/tex]
[tex]q = a_{2} / a_{1}[/tex]
[tex]q = a_{3}/a_{2}[/tex]
[tex]a_{2}/a_{1}=a_{3}/a_{2}[/tex]
[tex]a_{2}*a_{2} = a_{3}*a_{1}[/tex]
[tex](a_{2})^{2} = a_{1}*a_{3}[/tex]
[tex]n^{2} = m*(-8)[/tex]
[tex]n^{2} = - 8m[/tex]
Como m = 2 - n:
[tex]n^{2} = - 8(2 - n)[/tex]
[tex]n^{2} = - 16 + 8n[/tex]
[tex]n^{2} - 8n + 16 = 0[/tex]
[tex]S = -\frac{b}{a} = -\frac{(-8)}{1} = 8[/tex]
[tex]P = \frac{c}{a} = \frac{16}{1} = 16[/tex]
Raízes: 2 números que quando somados dão 8 e quando multiplicados dão 16:
[tex]n' = n'' = 4[/tex]
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2.
[tex]P.G([1+x],[-2+x],[3+x])[/tex]
[tex](a_{2})^{2} = a_{1}*a_{3}[/tex]
[tex](-2+x)^{2} = (1+x)(3+x)[/tex]
[tex](-2)^{2} + 2*(-2)*x + x^{2} = 3 + x + 3x + x^{2}[/tex]
[tex]4 - 4x = 3 + 4x[/tex]
[tex]4 - 3 = 4x + 4x[/tex]
[tex]1 = 8x[/tex]
[tex]1/8 = x[/tex]
[tex]a_{1} = 1 + x = 1 + (1/8) = (8 + 1)/8 = 9/8[/tex]
[tex]a_{2} = -2+x = -2 + (1/8) = (- 16 + 1)/8 = - 15/8[/tex]
[tex]q = a_{2} / a_{1}[/tex]
[tex]q = (-15/8)/(9/8)[/tex]
[tex]q = (-15/8)*(8/9)[/tex]
[tex]q = -15/9[/tex]
