Задача Найди сумму первых шестна ... если a11=7,3 и a23=33,7 (на арифметическую прогрессию)

Вы ввели

Найди сумму первых шестнадцати членов арифметической прогрессии, если a11=7,3 и a23=33,7

Найдено в тексте задачи:

Первый член: a1 = ?

n-член an (n = 22 + 1 = 23)

Разность: d = ?

Другие члены: a11 = (73/10)
a23 = (337/10)

Пример: ?

Найти члены от 1 до 23

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_23 - a_11
d = -----------
         12    

$$d = \frac{- a_{11} + a_{23}}{12}$$

             a_23 - a_11   
a_1 = a_23 - -----------*21
                  12       

$$a_{1} = a_{23} - 21 \frac{- a_{11} + a_{23}}{12}$$

    337   73
    --- - --
     10   10
d = --------
       12   

$$d = \frac{- \frac{73}{10} + \frac{337}{10}}{12}$$

            337   73   
            --- - --   
      337    10   10   
a_1 = --- - --------*22
       10      12      

$$a_{1} = - 22 \frac{- \frac{73}{10} + \frac{337}{10}}{12} + \frac{337}{10}$$

d = 11/5

$$d = \frac{11}{5}$$

      -147 
a_1 = -----
        10 

$$a_{1} = - \frac{147}{10}$$

Пример

...

Расширенный пример:

-147/10; -25/2; -103/10; -81/10; -59/10; -37/10; -3/2; 7/10; 29/10; 51/10; 73/10; 19/2; 117/10; 139/10; 161/10; 183/10; 41/2; 227/10; 249/10; 271/10; 293/10; 63/2; 337/10...

     -147 
a1 = -----
       10 

$$a_{1} = - \frac{147}{10}$$

a2 = -25/2

$$a_{2} = - \frac{25}{2}$$

     -103 
a3 = -----
       10 

$$a_{3} = - \frac{103}{10}$$

     -81 
a4 = ----
      10 

$$a_{4} = - \frac{81}{10}$$

     -59 
a5 = ----
      10 

$$a_{5} = - \frac{59}{10}$$

     -37 
a6 = ----
      10 

$$a_{6} = - \frac{37}{10}$$

a7 = -3/2

$$a_{7} = - \frac{3}{2}$$

a8 = 7/10

$$a_{8} = \frac{7}{10}$$

     29
a9 = --
     10

$$a_{9} = \frac{29}{10}$$

      51
a10 = --
      10

$$a_{10} = \frac{51}{10}$$

      73
a11 = --
      10

$$a_{11} = \frac{73}{10}$$

a12 = 19/2

$$a_{12} = \frac{19}{2}$$

      117
a13 = ---
       10

$$a_{13} = \frac{117}{10}$$

      139
a14 = ---
       10

$$a_{14} = \frac{139}{10}$$

      161
a15 = ---
       10

$$a_{15} = \frac{161}{10}$$

      183
a16 = ---
       10

$$a_{16} = \frac{183}{10}$$

a17 = 41/2

$$a_{17} = \frac{41}{2}$$

      227
a18 = ---
       10

$$a_{18} = \frac{227}{10}$$

      249
a19 = ---
       10

$$a_{19} = \frac{249}{10}$$

      271
a20 = ---
       10

$$a_{20} = \frac{271}{10}$$

      293
a21 = ---
       10

$$a_{21} = \frac{293}{10}$$

a22 = 63/2

$$a_{22} = \frac{63}{2}$$

      337
a23 = ---
       10

$$a_{23} = \frac{337}{10}$$

...

Разность

d = 11/5

$$d = \frac{11}{5}$$

Сумма

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

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

S23 = 437/2

$$S_{23} = \frac{437}{2}$$

Первый член

      -147 
a_1 = -----
        10 

$$a_{1} = - \frac{147}{10}$$

n-член

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

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

       337
a_23 = ---
        10

$$a_{23} = \frac{337}{10}$$