1)
![a_1=-4; d=-2-(-4)=2\\S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{10}=\frac{2*(-4)+2(10-1)}{2}*10= 10*5=50 a_1=-4; d=-2-(-4)=2\\S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{10}=\frac{2*(-4)+2(10-1)}{2}*10= 10*5=50](https://tex.z-dn.net/?f=a_1%3D-4%3B%20d%3D-2-%28-4%29%3D2%5C%5CS_n%3D%5Cfrac%7B2a_1%2Bd%28n-1%29%7D%7B2%7D%20%2An%5C%5CS_%7B10%7D%3D%5Cfrac%7B2%2A%28-4%29%2B2%2810-1%29%7D%7B2%7D%2A10%3D%2010%2A5%3D50)
2)
![a_1=-7; d=-5-(-7)=2\\a_n=a_1+d(n-1)\\a_{16}=-7+2(16-1)=23 a_1=-7; d=-5-(-7)=2\\a_n=a_1+d(n-1)\\a_{16}=-7+2(16-1)=23](https://tex.z-dn.net/?f=a_1%3D-7%3B%20d%3D-5-%28-7%29%3D2%5C%5Ca_n%3Da_1%2Bd%28n-1%29%5C%5Ca_%7B16%7D%3D-7%2B2%2816-1%29%3D23)
3)
Для решения задачи воспользуемся арифметической прогрессии
Здесь
![a_1=30, d=2 a_1=30, d=2](https://tex.z-dn.net/?f=a_1%3D30%2C%20d%3D2)
![a_n=a_1+d(n-1)\\a_n=30+2(n-1)\\a_n=30+2n-2\\a_n=28+2n a_n=a_1+d(n-1)\\a_n=30+2(n-1)\\a_n=30+2n-2\\a_n=28+2n](https://tex.z-dn.net/?f=a_n%3Da_1%2Bd%28n-1%29%5C%5Ca_n%3D30%2B2%28n-1%29%5C%5Ca_n%3D30%2B2n-2%5C%5Ca_n%3D28%2B2n)
4)
![a_1=-8,6, d=-8,4-(-8,6)=0,2\\a_n=a_1+d(n-1)\\-8,6+0,2(n-1)<0\\-8,6+0,2n-0,2<0\\0,2n<8,8\\n<44 a_1=-8,6, d=-8,4-(-8,6)=0,2\\a_n=a_1+d(n-1)\\-8,6+0,2(n-1)<0\\-8,6+0,2n-0,2<0\\0,2n<8,8\\n<44](https://tex.z-dn.net/?f=a_1%3D-8%2C6%2C%20d%3D-8%2C4-%28-8%2C6%29%3D0%2C2%5C%5Ca_n%3Da_1%2Bd%28n-1%29%5C%5C-8%2C6%2B0%2C2%28n-1%29%3C0%5C%5C-8%2C6%2B0%2C2n-0%2C2%3C0%5C%5C0%2C2n%3C8%2C8%5C%5Cn%3C44)
, значит прогрессия имеет 43 отрицательных члена. Найдем сумму первых 43 членов данной прогрессии
![S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{43}=\frac{2*(-8,6)+0,2(43-1)}{2} *43=-189,2 S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{43}=\frac{2*(-8,6)+0,2(43-1)}{2} *43=-189,2](https://tex.z-dn.net/?f=S_n%3D%5Cfrac%7B2a_1%2Bd%28n-1%29%7D%7B2%7D%20%2An%5C%5CS_%7B43%7D%3D%5Cfrac%7B2%2A%28-8%2C6%29%2B0%2C2%2843-1%29%7D%7B2%7D%20%2A43%3D-189%2C2)
5)
![a_1=-2, d=3\\a_n=a_1+d(n-1)\\a_4=-2+3(4-1)=7 a_1=-2, d=3\\a_n=a_1+d(n-1)\\a_4=-2+3(4-1)=7](https://tex.z-dn.net/?f=a_1%3D-2%2C%20d%3D3%5C%5Ca_n%3Da_1%2Bd%28n-1%29%5C%5Ca_4%3D-2%2B3%284-1%29%3D7)
6)
, значит d=0,9
![a_1=-3,1 a_1=-3,1](https://tex.z-dn.net/?f=a_1%3D-3%2C1)
![S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{19}=\frac{2*(-3,1)+0,9(19-1)}{2} *19=95 S_n=\frac{2a_1+d(n-1)}{2} *n\\S_{19}=\frac{2*(-3,1)+0,9(19-1)}{2} *19=95](https://tex.z-dn.net/?f=S_n%3D%5Cfrac%7B2a_1%2Bd%28n-1%29%7D%7B2%7D%20%2An%5C%5CS_%7B19%7D%3D%5Cfrac%7B2%2A%28-3%2C1%29%2B0%2C9%2819-1%29%7D%7B2%7D%20%2A19%3D95)
7)
⇒
, значит
![19=a_1+9d\\44=a_1+14d 19=a_1+9d\\44=a_1+14d](https://tex.z-dn.net/?f=19%3Da_1%2B9d%5C%5C44%3Da_1%2B14d)
Вычитаем из второго уравнения первое, получим
, отсюда
![d=5 d=5](https://tex.z-dn.net/?f=d%3D5)