[tex]a_{6}=5\\ a_{9}=17\\a_{1}=?,r=?, S_{20}=?, a_{n}=?\\a_{6}+a_{9}= a_{6}+a_{6}+3r=5+17\\2a_{6}+3r=22\\10+3r=22\\\3r=12\\r=4\\a_{6}=a_{1}+5r=5\\a_{1}=5-5*4\\a_{1}=-15\\S_{20}=\frac{2a_{1}+(20-1)r}{2} *20=\frac{-30 + 19*4}{2} *20= 460\\a_{n} = a_{1} + (n-1)r = -15+ (n-1)*4=-15+4n-4=4n-19[/tex]
Odpowiedź:
[tex]a_{6}=5\ \ ,\ \ a_{9}=17\\\\\\Obliczymy\ \ r\'o\.znice\ \ ciagu\\\\r=\dfrac{a_{9}-a_{6}}{9-6}=\dfrac{17-5}{3}=\dfrac{12}{3}=4\\\\\\Obliczymy\ \ wyraz\ \ pierwszy\ \ tego\ \ ciagu\\\\a_{n}=a_{1}+(n-1)\cdot r\\\\a_{6}=a_{1}+(6-1)\cdot4\\\\5=a_{1}+5\cdot4\\\\5=a_{1}+20\\\\5-20=a_{1}\\\\-15=a_{1}\\\\a_{1}=-15[/tex]
[tex]Obliczamy\ \ sume\ \ wyraz\'ow\ \ ciagu\ \ arytmetycznego\ \ korzystajac\ \ ze\ \ wzoru\\\\S_{n}=\frac{2a_{1}+(n-1)\cdot r}{2}\cdot n\\\\S_{20}=\frac{2\cdot(-15)+(20-1)\cdot4}{2}\cdot20\\\\S_{20}=\frac{-30+19\cdot4}{2}\cdot20\\\\S_{20}=\frac{-30+76}{\not2_{1} }\cdot\not20^1^0\\\\S_{20}=46\cdot10=460\\\\\\Wz\'or\ \ og\'olny\ \ ciagu\ \ a_{n}=a_{1}+(n-1)\cdot r\\\\a_{n}=-15+(n-1)\cdot4\\\\a_{n}=-15+4n-4\\\\a_{n}=4n-19[/tex]
