Respuesta :
Answer:
U₁₀ = {17, -25}
Step-by-step explanation:
Arithmetic Sequence
[tex]\boxed{U_n=a+(n-1)d}[/tex]
- [tex]U_1=a+(1-1)d=a[/tex]
- [tex]U_3=a+(3-1)d=a+2d[/tex]
- [tex]U_5=a+(5-1)d=a+4d[/tex]
[tex]a_1+a_3+a_5=-12[/tex]
[tex]a+(a+2d)+(a+4d)=-12[/tex]
[tex]3a+6d=-12[/tex]
[tex]a+2d=-4\ ...\ [1][/tex]
[tex]2d=-4-a[/tex]
[tex]d=-2-\frac{a}{2}\ ...\ [2][/tex]
[tex]a_1\cdot a_3\cdot a_5=80[/tex]
[tex]a(a+2d)(a+4d)=80[/tex]
[tex]a(-4)(a+4d)=80[/tex] (Refer to [1], we can replace a+2d with -4)
[tex]a(a+4d)=80\div(-4)[/tex]
[tex]a(a+4(-2-\frac{a}{2} ))=-20[/tex] (Refer to [2])
[tex]a(a-2-2a)=-20[/tex]
[tex]a(-a-2)=-20[/tex]
[tex]a^2+2a-20=0[/tex]
[tex](a+10)(a-2)=0[/tex]
[tex]\bf a_1=-10[/tex] → [tex]d_1=-2-\frac{-10}{2}[/tex]
[tex]\bf d_1=3[/tex]
[tex]\bf a_2=2[/tex] → [tex]d_2=-2-\frac{2}{2}[/tex]
[tex]\bf d_2=-3[/tex]
[tex]\boxed{U_{10}=a+(10-1)d}[/tex]
(1) a = -10 and d = 3
[tex]U_{10}=-10+(10-1)(3)[/tex]
[tex]=17[/tex]
(2) a = 2 and d = -3
[tex]U_{10}=2+(10-1)(-3)[/tex]
[tex]=-25[/tex]
