1.
[tex]a_{n} = 2n^{2}-6n\\\\a_4 = 2\cdot4^{2}-6\cdot4 = 2\cdot16 - 24 = 32 - 24 = 8\\\\a_4 = 8[/tex]
2.
[tex]a_{n} = 6n^{2}\\\\6n^{2} = 150 \ \ /:6\\\\n^{2} = 25 \in \ N+\\\\n = 5\\\\a_5 = 150[/tex]
3.
[tex]a_3 = 7 = a_1 + 2r\\a_4 = 22 = a_1 + 3r\\a_1 = ?\\r = ?\\\\a_1 + 2r = 7 \ \ /\cdot(-3)\\a_1 + 3r = 22 \ \ /\cdot2\\\\-3a_1 - 6r = -21\\2a_1 + 6r = 44\\-------- \ \ \ +\\-a_1 = 23\\a_1 = -23\\\\r = a_4-a_3 = 22-7 = 15\\r = 15[/tex]
4.
[tex]a_1 = 6\\r = 2\\n = 24\\S_{24} = ?\\\\S_{n} = \frac{a_1+(n-1)\cdot r}{2}\cdot n\\\\S_{24} = \frac{6+(24-1)\cdot2}{2}\cdot24 = \frac{6+23\cdot2}{2}\cdot24} = 52\cdot12 = 624\\\\S_{24} = 624[/tex]
5.
Liczby te tworzą cią arytmetyczny, w którym:
[tex]a_1 = 100\\r = 2\\a_{n} = 998\\\\a_{n} = a_1 + (n-1)\cdot r\\\\998 = 100+(n-1)\cdot2\\\\100+2n-2 = 998\\\\2n = 900 \ \ /:2\\\\n = 450[/tex]
[tex]S_{n} = \frac{a_1 + a_{n}}{2}\cdot n\\\\S_{450} = \frac{100+998}{2}\cdot450 + 1098\cdot225 = 247050\\\\S_{45} = 247050[/tex]
6.
[tex]a_{n} = n^{2}-6n+5\\\\n^{2}-6n+5<0\\\\\Delta = (-6)^{2}-4\cdot5 = 36-20 = 16\\\\\sqrt{\Delta} = \sqrt{16} = 4\\\\n_1 = \frac{6-4}{2} = \frac{2}{2} = 1\\\\n_2 = \frac{6+4}{2} = \frac{10}{2} = 5\\\\n\in(1,5)\\\\n\in\{2,3,4\}[/tex]
Odp. Ten ciąg ma trzy wyrazy ujemne: a₂, a₃, a₄.
7.
[tex]a_{n} = 4n + 2\\\\a_{n+1} = 4(n+1)+2 = 4n + 4 + 2 = 4n+6\\\\r = a_{n+1}-a_{n} = 4n+6 - (4n+2) = 4n+6-4n-2 = 4[/tex]
Odp. Różnica ciągu r jest stała, więc ciąg jest ciągiem arytmetycznym.
