Question

Ciąg geometryczny
a1=√8
q=-√2
n=8
Oblicz 8 wyraz ciągu i sumę

Answer 1

Odpowiedź:

[tex]a_1=\sqrt{8}[/tex]

[tex]q=-\sqrt{2}[/tex]

[tex]n=8[/tex]

[tex]a_n=a_1*q^{n-1}\\a_8=\sqrt{8} *(-\sqrt{2}) ^{7}\\a_8=\sqrt{2^3} *(-\sqrt{2^7}) \\a_8=-\sqrt{2^{3+7}}\\a_8=-\sqrt{2^{10}}\\a_8=-(2^{10})^{\frac{1}{2} }\\a_8=-2^5\\a_8=-32[/tex]

[tex]S_n=a_1*\frac{1-q^n}{1-q} \\S_8=\sqrt{8}*\frac{1-(-\sqrt{2}) ^8}{1-(-\sqrt{2}) } \\S_8=\sqrt{8}*\frac{1-({-2}^{\frac{1}{2} })^8}{1+\sqrt{2} } \\S_8=\sqrt{2^3}*\frac{1-2^4}{1+\sqrt{2} } *\frac{1-\sqrt{2} }{1-\sqrt{2}} \\S_8=\sqrt{2*2^2}*\frac{(1-16)(1-\sqrt{2})}{1-2 } \\S_8=2\sqrt{2}*\frac{(-15)(1-\sqrt{2})}{-1 } \\S_8=2\sqrt{2}*15(1-\sqrt{2})\\S_8=30\sqrt{2}(1-\sqrt{2})\\S_8=30\sqrt{2}-30\sqrt{2}*\sqrt{2}\\S_8=30\sqrt{2}-30\sqrt{2}^2\\S_8=30\sqrt{2}-30*2\\S_8=30\sqrt{2}-60[/tex]