Задача вычислите сумму восьми пе ... сии an,если a1 =8 и a8=13 (на арифметическую прогрессию)

Вы ввели

вычислите сумму восьми первых челенов арифметической прогрессии an,если a1 =8 и a18=13

Найдено в тексте задачи:

Первый член: a1 = 8

n-член an (n = 17 + 1 = 18)

Разность: d = ?

Другие члены: a1 = 8
a18 = 13

Пример: ?

Найти члены от 1 до 18

Решение

    a_n - a_k
d = ---------
      n - k  

$$d = \frac{- a_{k} + a_{n}}{- k + n}$$

a_1 = a_n + d*(-1 + n)

$$a_{1} = a_{n} + d \left(n - 1\right)$$

            (-1 + n)*(a_n - a_k)
a_1 = a_n - --------------------
                   n - k        

$$a_{1} = a_{n} - \frac{\left(- a_{k} + a_{n}\right) \left(n - 1\right)}{- k + n}$$

    a_18 - a_1
d = ----------
        17    

$$d = \frac{- a_{1} + a_{18}}{17}$$

             a_18 - a_1   
a_1 = a_18 - ----------*16
                 17       

$$a_{1} = a_{18} - \frac{- a_{1} + a_{18}}{17} \cdot 16$$

    13 - 8
d = ------
      17  

$$d = \frac{-8 + 13}{17}$$

           13 - 8   
a_1 = 13 - ------*17
             17     

$$a_{1} = \left(-1\right) \frac{-8 + 13}{17} \cdot 17 + 13$$

d = 5/17

$$d = \frac{5}{17}$$

a_1 = 8

$$a_{1} = 8$$

Первый член

a_1 = 8

$$a_{1} = 8$$

Пример

...

Расширенный пример:

8; 141/17; 146/17; 151/17; 156/17; 161/17; 166/17; 171/17; 176/17; 181/17; 186/17; 191/17; 196/17; 201/17; 206/17; 211/17; 216/17; 13...

a1 = 8

$$a_{1} = 8$$

     141
a2 = ---
      17

$$a_{2} = \frac{141}{17}$$

     146
a3 = ---
      17

$$a_{3} = \frac{146}{17}$$

     151
a4 = ---
      17

$$a_{4} = \frac{151}{17}$$

     156
a5 = ---
      17

$$a_{5} = \frac{156}{17}$$

     161
a6 = ---
      17

$$a_{6} = \frac{161}{17}$$

     166
a7 = ---
      17

$$a_{7} = \frac{166}{17}$$

     171
a8 = ---
      17

$$a_{8} = \frac{171}{17}$$

     176
a9 = ---
      17

$$a_{9} = \frac{176}{17}$$

      181
a10 = ---
       17

$$a_{10} = \frac{181}{17}$$

      186
a11 = ---
       17

$$a_{11} = \frac{186}{17}$$

      191
a12 = ---
       17

$$a_{12} = \frac{191}{17}$$

      196
a13 = ---
       17

$$a_{13} = \frac{196}{17}$$

      201
a14 = ---
       17

$$a_{14} = \frac{201}{17}$$

      206
a15 = ---
       17

$$a_{15} = \frac{206}{17}$$

      211
a16 = ---
       17

$$a_{16} = \frac{211}{17}$$

      216
a17 = ---
       17

$$a_{17} = \frac{216}{17}$$

a18 = 13

$$a_{18} = 13$$

...

Разность

d = 5/17

$$d = \frac{5}{17}$$

Сумма

    n*(a_1 + a_n)
S = -------------
          2      

$$S = \frac{n \left(a_{1} + a_{n}\right)}{2}$$

Сумма восемнадцати членов

      18*(8 + 13)
S18 = -----------
           2     

$$S_{18} = \frac{18 \cdot \left(8 + 13\right)}{2}$$

S18 = 189

$$S_{18} = 189$$

n-член

Восемнадцатый член

a_n = a_1 + d*(-1 + n)

$$a_{n} = a_{1} + d \left(n - 1\right)$$

a_18 = 13

$$a_{18} = 13$$