[tex]a_{n} = -2n+3\\\\a)\\\\r = a_{n+1}-a_{n} = -2(n+1)+3-(-2n+3) =-2n-2+3+2n-3=-2[/tex]
Odp. Tak, dany ciąg jest arytmetyczny.
[tex]b)\\r = -2\\a_1 = -2\cdot1+3 =-2+3 = 1\\\\S_{n} = \frac{2a_1+(n-1)r}{2}\cdot n\\\\\\S_{45} = \frac{2\cdot1+(45-1)\cdot(-2)}{2}\cdot45=\frac{2+44\cdot(-2)}{2}\cdot45=\frac{2-88}{2}\cdot45 = \frac{-86}{2}\cdot45=-43\cdot45 = -1935[/tex]
[tex]S_{50} = \frac{2\cdot1+(50-1)\cdot(-2)}{2}\cdot50 =\frac{2+49\cdot(-2)}{2}\cdot50} =\frac{2-98}{2}\cdot50 = \frac{-96}{2}\cdot50 = -96\cdot25 = -2400[/tex]
[tex]S_{45}-S_{50} = -1935-(-2400) = -1935+2400 = -465[/tex]
