Obliczmy różnicę
[tex]a_{12} + 3r = a_{15} \\ 10 + 3r = 12 \\ 3r = 2 | \div 3 \\ r = \frac{2}{3} [/tex]
Pierwszy wyraz
[tex]a_{1} + 14r = a_{15} \\ a_{1} + 14 \times \frac{2}{3} = 12 \\ a_{1} + \frac{28}{3} = 12 \\ a_{1} = 12 - \frac{28}{3} \\ a_{1} = \frac{36}{3} - \frac{28}{3} \\ a_{1} = \frac{8}{3} = 2 \frac{2}{3} [/tex]
Zatem szesnasty wyraz
[tex]a_{15} + r = a_{16} \\ 12 + \frac{2}{3} = a_{16} \\ \boxed{ a_{16} = 12 \frac{2}{3} }[/tex]
Dwudziesty wyraz
[tex]a_{15} + 5r = a_{20} \\ 12 + 5 \times \frac{2}{3} = a_{20} \\ 12 + \frac{10}{3} = a_{20} \\ a_{20} = 12 + 3 \frac{1}{3} \\ a_{20} = 15 \frac{1}{3} [/tex]
Suma 20 pierwszych wyrazów
[tex]S_{20} = \frac{a_1 + a_{20}}{2} \times 20 \\S_{20} = \frac{2 \frac{2}{3} + 15 \frac{1}{3} }{2} \times 20 \\ S_{20} = \frac{18}{2} \times 20 \\ S_{20} = 9 \times 20 \\ \boxed{ S_{20} = 180}[/tex]