На дне озера может быть много вещей полезных для ученых и биологов , и археологов. Например в Северной Америке произвели исследователское погружение на дно озера Вашингтон и тма были обнаружены скелеты древнего человека динозавра и мамонта
Дано:
![a_n - arithmetic \: \: \: progression](https://tex.z-dn.net/?f=a_n+-+arithmetic+%5C%3A+%5C%3A+%5C%3A+progression)
![a_8=14](https://tex.z-dn.net/?f=a_8%3D14+)
![a_{14}=8](https://tex.z-dn.net/?f=a_%7B14%7D%3D8)
Найти:
![S_{21} - ?](https://tex.z-dn.net/?f=+S_%7B21%7D+-+%3F)
Решение:
Воспользуемся формулой
![a_n = a_1+ d \times (n - 1)](https://tex.z-dn.net/?f=a_n+%3D+a_1%2B+d+%5Ctimes+%28n+-+1%29+)
И составим систему уравнений
![|a_8 = a_1+ 7d \\ |a_{14} = a_1+ 15d \\](https://tex.z-dn.net/?f=+%7Ca_8+%3D+a_1%2B+7d+%5C%5C+%7Ca_%7B14%7D+%3D+a_1%2B+15d+%5C%5C+)
Отсюда
![- 8d = 6 ; \\d = - \frac{6}{8} ; \\ d = - \frac{3}{4} ;](https://tex.z-dn.net/?f=+-+8d+%3D+6+%3B+%5C%5Cd+%3D+-+%5Cfrac%7B6%7D%7B8%7D+%3B+%5C%5C+d+%3D+-+%5Cfrac%7B3%7D%7B4%7D+%3B)
Тогда
![а_{8} = а_{1} - \frac{3}{4} \times 7 ; \\ а_{1} =14 + 5.25 ; \\ а_{1} = 19.25;](https://tex.z-dn.net/?f=%D0%B0_%7B8%7D+%3D+%D0%B0_%7B1%7D+-+%5Cfrac%7B3%7D%7B4%7D+%5Ctimes+7+%3B+%5C%5C+%D0%B0_%7B1%7D+%3D14+%2B+5.25+%3B+%5C%5C+%D0%B0_%7B1%7D+%3D+19.25%3B)
Используем 2 и 3 формулы,
![< 2> \: \: \: а_{21} = а_{1} + d \times (n - 1) ; \\ < 3 > \: \: \: S_{n} = \frac{а_{1} + а_{n} }{2} \times n](https://tex.z-dn.net/?f=+%3C+2%3E+%5C%3A+%5C%3A+%5C%3A+%D0%B0_%7B21%7D+%3D+%D0%B0_%7B1%7D+%2B+d+%5Ctimes+%28n+-+1%29+%3B+%5C%5C+%3C+3+%3E+%5C%3A+%5C%3A+%5C%3A+S_%7Bn%7D+%3D+%5Cfrac%7B%D0%B0_%7B1%7D+%2B+%D0%B0_%7Bn%7D+%7D%7B2%7D+%5Ctimes+n)
![< 2 > \: \: \: а_{21} = 19 .25 + 20 \times ( - \frac{3}{4} ) = 4.25 \\ < 3 > S_{21} = \frac{19.25 +4.25 }{2} \times 20 = 235](https://tex.z-dn.net/?f=+%3C+2+%3E+%5C%3A+%5C%3A+%5C%3A+%D0%B0_%7B21%7D+%3D+19+.25+%2B+20+%5Ctimes+%28+-+%5Cfrac%7B3%7D%7B4%7D+%29+%3D+4.25+%5C%5C+%3C+3+%3E+S_%7B21%7D+%3D+%5Cfrac%7B19.25+%2B4.25+%7D%7B2%7D+%5Ctimes+20+%3D+235)
Ответ:
![S_{21} = 235](https://tex.z-dn.net/?f=S_%7B21%7D+%3D+235)