Задача найдите разность арифмети ... ессии, если а1=53 а20=-33 (на арифметическую прогрессию)

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найдите разность арифметической прогрессии, если а1=53 а20=-33

Найдено в тексте задачи:

Первый член: a1 = 53

n-член an (n = 19 + 1 = 20)

Разность: d = ?

Другие члены: a1 = 53
a20 = -33

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       

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

    -33 - 53
d = --------
       19   

$$d = \frac{-53 - 33}{19}$$

            -33 - 53   
a_1 = -33 - --------*19
               19      

$$a_{1} = -33 - \frac{-53 - 33}{19} \cdot 19$$

    -86 
d = ----
     19 

$$d = - \frac{86}{19}$$

a_1 = 53

$$a_{1} = 53$$

Пример

...

Расширенный пример:

53; 921/19; 835/19; 749/19; 663/19; 577/19; 491/19; 405/19; 319/19; 233/19; 147/19; 61/19; -25/19; -111/19; -197/19; -283/19; -369/19; -455/19; -541/19; -33...

a1 = 53

$$a_{1} = 53$$

     921
a2 = ---
      19

$$a_{2} = \frac{921}{19}$$

     835
a3 = ---
      19

$$a_{3} = \frac{835}{19}$$

     749
a4 = ---
      19

$$a_{4} = \frac{749}{19}$$

     663
a5 = ---
      19

$$a_{5} = \frac{663}{19}$$

     577
a6 = ---
      19

$$a_{6} = \frac{577}{19}$$

     491
a7 = ---
      19

$$a_{7} = \frac{491}{19}$$

     405
a8 = ---
      19

$$a_{8} = \frac{405}{19}$$

     319
a9 = ---
      19

$$a_{9} = \frac{319}{19}$$

      233
a10 = ---
       19

$$a_{10} = \frac{233}{19}$$

      147
a11 = ---
       19

$$a_{11} = \frac{147}{19}$$

      61
a12 = --
      19

$$a_{12} = \frac{61}{19}$$

      -25 
a13 = ----
       19 

$$a_{13} = - \frac{25}{19}$$

      -111 
a14 = -----
        19 

$$a_{14} = - \frac{111}{19}$$

      -197 
a15 = -----
        19 

$$a_{15} = - \frac{197}{19}$$

      -283 
a16 = -----
        19 

$$a_{16} = - \frac{283}{19}$$

      -369 
a17 = -----
        19 

$$a_{17} = - \frac{369}{19}$$

      -455 
a18 = -----
        19 

$$a_{18} = - \frac{455}{19}$$

      -541 
a19 = -----
        19 

$$a_{19} = - \frac{541}{19}$$

a20 = -33

$$a_{20} = -33$$

...

Первый член

a_1 = 53

$$a_{1} = 53$$

Разность

    -86 
d = ----
     19 

$$d = - \frac{86}{19}$$

n-член

Двадцатый член

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

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

a_20 = -33

$$a_{20} = -33$$

Сумма

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

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

Сумма двадцати членов

      20*(53 - 33)
S20 = ------------
           2      

$$S_{20} = \frac{20 \left(-33 + 53\right)}{2}$$

S20 = 200

$$S_{20} = 200$$