Задача a1=10 a20=120 S20=? (на арифметическую прогрессию)

Вы ввели

a1=10 a20=120 S20=?

Найдено в тексте задачи:

Первый член: a1 = 10

n-член an (n = 19 + 1 = 20)

Разность: d = ?

Другие члены: a1 = 10
a20 = 120

Пример: ?

Найти члены от 1 до 20

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_20 - a_1
d = ----------
        19    

$$d = \frac{- a_{1} + a_{20}}{19}$$

             a_20 - a_1   
a_1 = a_20 - ----------*18
                 19       

$$a_{1} = a_{20} - 18 \frac{- a_{1} + a_{20}}{19}$$

    120 - 10
d = --------
       19   

$$d = \frac{-10 + 120}{19}$$

            120 - 10   
a_1 = 120 - --------*19
               19      

$$a_{1} = - 19 \frac{-10 + 120}{19} + 120$$

    110
d = ---
     19

$$d = \frac{110}{19}$$

a_1 = 10

$$a_{1} = 10$$

Первый член

a_1 = 10

$$a_{1} = 10$$

Пример

...

Расширенный пример:

10; 300/19; 410/19; 520/19; 630/19; 740/19; 850/19; 960/19; 1070/19; 1180/19; 1290/19; 1400/19; 1510/19; 1620/19; 1730/19; 1840/19; 1950/19; 2060/19; 2170/19; 120...

a1 = 10

$$a_{1} = 10$$

     300
a2 = ---
      19

$$a_{2} = \frac{300}{19}$$

     410
a3 = ---
      19

$$a_{3} = \frac{410}{19}$$

     520
a4 = ---
      19

$$a_{4} = \frac{520}{19}$$

     630
a5 = ---
      19

$$a_{5} = \frac{630}{19}$$

     740
a6 = ---
      19

$$a_{6} = \frac{740}{19}$$

     850
a7 = ---
      19

$$a_{7} = \frac{850}{19}$$

     960
a8 = ---
      19

$$a_{8} = \frac{960}{19}$$

     1070
a9 = ----
      19 

$$a_{9} = \frac{1070}{19}$$

      1180
a10 = ----
       19 

$$a_{10} = \frac{1180}{19}$$

      1290
a11 = ----
       19 

$$a_{11} = \frac{1290}{19}$$

      1400
a12 = ----
       19 

$$a_{12} = \frac{1400}{19}$$

      1510
a13 = ----
       19 

$$a_{13} = \frac{1510}{19}$$

      1620
a14 = ----
       19 

$$a_{14} = \frac{1620}{19}$$

      1730
a15 = ----
       19 

$$a_{15} = \frac{1730}{19}$$

      1840
a16 = ----
       19 

$$a_{16} = \frac{1840}{19}$$

      1950
a17 = ----
       19 

$$a_{17} = \frac{1950}{19}$$

      2060
a18 = ----
       19 

$$a_{18} = \frac{2060}{19}$$

      2170
a19 = ----
       19 

$$a_{19} = \frac{2170}{19}$$

a20 = 120

$$a_{20} = 120$$

...

Разность

    110
d = ---
     19

$$d = \frac{110}{19}$$

Сумма

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

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

Сумма двадцати членов

S20 = 1300

$$S_{20} = 1300$$

n-член

Двадцатый член

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

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

a_20 = 120

$$a_{20} = 120$$