[tex]zad.1\\\\a_{n} =(-1)^{n} \cdot 2n\\\\a_{1} =(-1)^{1} \cdot 2\cdot 1=(-1)\cdot 2=-2\\\\a_{2} =(-1)^{2} \cdot 2\cdot 2=1\cdot 4=4\\\\a_{3} =(-1)^{3} \cdot 2\cdot 3=(-1)\cdot 6=-6[/tex]
zad.2
[tex]a_{n} =a_{1} +(n-1)\cdot r[/tex] - wzór na n-ty wyraz ciągu artymetycznego o pierwszym wyrazie a₁ i różnicy r.
[tex]a_{2} =-3~~\land~~a_{2}=a_{1} +(2-1)\cdot r ~~\Rightarrow~~a_{1} +r=-3\\\\a_{1} +r=-3~~\land~~r=5~~\Rightarrow ~~a_{1} +5=-3\\\\a_{1} +5=-3\\\\a_{1} =-3-5\\\\a_{1} =-8\\\\a_{n} =a_{1} +(n-1)\cdot r ~~\land~~a_{1} =-8~~\land~~r=5~~\Rightarrow~~a_{n} =-8+(n-1)\cdot 5\\\\a_{n} =-8+(n-1)\cdot 5\\\\a_{n} =-8+5n-5\\\\a_{n} =5n-13[/tex]
Odp: Szukany wzór ogólny ciągu arytmetycznego : aₙ = 5n - 13.
zad.3
[tex]a_{n} =a_{1} \cdot q^{n-1} ~~dla~~n\geq 2[/tex] wzór na n-ty wyraz ciągu geometrycznego o pierwszym wyrazie a₁ i ilorazie q.
[tex]a_{3} =125~~\land~~a_{3} =a_{1} \cdot q^{3-1} ~~\Rightarrow~~a_{1} \cdot q^{2}=125\\\\a_{1} \cdot q^{2}=125~~\land~~q=-5~~\Rightarrow~~a_{1} \cdot (-5)^{2}=125\\\\a_{1} \cdot (-5)^{2}=125\\\\a_{1} \cdot 25=125~~\mid \div 25\\\\a_{1} =5[/tex]
[tex]a_{n} =a_{1} \cdot q^{n-1} ~~\land~~q=-5~~\land~~a_{1} =5\\\\a_{n} =5 \cdot (-5)^{n-1}\\\\a_{n} =5 \cdot \dfrac{(-5)^{n} }{(-5)^{1} } \\\\a_{n} =5 \cdot \dfrac{(-5)^{n} }{(-5) } \\\\a_{n} =-(-5)^{n}[/tex]
Odp: Szukany wzór ciągu geometrycznego : aₙ = -(-5)ⁿ.
