Zadanie 1.
[tex]a_5 = a_1 + 4r = 12[/tex]
[tex]a_6 = a_1 + 5r = 7[/tex]
[tex]a_6 - a_5 = a1 + 5r - a1 - 4r = r = 7 - 12 = -5[/tex]
r = -5
[tex]a_1 - 20 = 12\\a_1 = 32[/tex]
Wzór ciągu arytmetycznego na n-ty wyraz ma postać:
[tex]a_n = a_1 + r(n-1)[/tex]
Podstawiamy:
[tex]a_n = 32 - 5(n-1) = 32 - 5n + 5 = 37 - 5n[/tex]
[tex]S_{12} = \frac{2a_1 + (n-1)r}{2} * n = \frac{64 - 5 * 11}{2} * 12 = 54[/tex]
Zadanie 2.
[tex]\frac{a_5}{a_4} = -\frac{1}{2} = q[/tex]
[tex]a_4 = a_1 * q^3 = -\frac{a_1}{8} = 12[/tex]
[tex]a_1 = -96[/tex]
[tex]S_5 = a_1*\frac{1-q^n}{1-q} = -96 * \frac{1+\frac{1}{32} }{1+\frac{1}{2} }[/tex]= [tex]-96 * \frac{33}{32} * \frac{2}{3} = -66[/tex]
Zadanie 3.
Dla ciągu geometrycznego istnieje zależność:
[tex]a_n^2 = a_{n-1}*a_{n+1}[/tex]
Zatem:
36 = x*(3x+3)
36 = 3x² + 3x
x² + x - 12 = 0
(x-3)(x+4) = 0
x = 3 v x = -4
x∈{-4, 3}
Zadanie 4.
n² - n - 2 = 0
(n-2)(n+1) = 0
n = 2 v n = -1
n∈{-1, 2}
Pozdrawiam
