Задача Найдите первый член арифм ... an), если a37=-152 a43=43 (на арифметическую прогрессию)

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Найдите первый член арифметической прогрессии (an), если a37=-152 a43=43

Найдено в тексте задачи:

Первый член: a1 = ?

n-член an (n = 42 + 1 = 43)

Разность: d = ?

Другие члены: a37 = -152
a43 = 43

Пример: ?

Найти члены от 1 до 43

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_43 - a_37
d = -----------
         6     

$$d = \frac{- a_{37} + a_{43}}{6}$$

             a_43 - a_37   
a_1 = a_43 - -----------*41
                  6        

$$a_{1} = a_{43} - 41 \frac{- a_{37} + a_{43}}{6}$$

    43 + 152
d = --------
       6    

$$d = \frac{43 + 152}{6}$$

           43 + 152   
a_1 = 43 - --------*42
              6       

$$a_{1} = - 42 \frac{43 + 152}{6} + 43$$

d = 65/2

$$d = \frac{65}{2}$$

a_1 = -1322

$$a_{1} = -1322$$

Первый член

a_1 = -1322

$$a_{1} = -1322$$

Разность

d = 65/2

$$d = \frac{65}{2}$$

n-член

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

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

a_43 = 43

$$a_{43} = 43$$

Пример

...

Расширенный пример:

-1322; -2579/2; -1257; -2449/2; -1192; -2319/2; -1127; -2189/2; -1062; -2059/2; -997; -1929/2; -932; -1799/2; -867; -1669/2; -802; -1539/2; -737; -1409/2; -672; -1279/2; -607; -1149/2; -542; -1019/2; -477; -889/2; -412; -759/2; -347; -629/2; -282; -499/2; -217; -369/2; -152; -239/2; -87; -109/2; -22; 21/2; 43...

a1 = -1322

$$a_{1} = -1322$$

a2 = -2579/2

$$a_{2} = - \frac{2579}{2}$$

a3 = -1257

$$a_{3} = -1257$$

a4 = -2449/2

$$a_{4} = - \frac{2449}{2}$$

a5 = -1192

$$a_{5} = -1192$$

a6 = -2319/2

$$a_{6} = - \frac{2319}{2}$$

a7 = -1127

$$a_{7} = -1127$$

a8 = -2189/2

$$a_{8} = - \frac{2189}{2}$$

a9 = -1062

$$a_{9} = -1062$$

a10 = -2059/2

$$a_{10} = - \frac{2059}{2}$$

a11 = -997

$$a_{11} = -997$$

a12 = -1929/2

$$a_{12} = - \frac{1929}{2}$$

a13 = -932

$$a_{13} = -932$$

a14 = -1799/2

$$a_{14} = - \frac{1799}{2}$$

a15 = -867

$$a_{15} = -867$$

a16 = -1669/2

$$a_{16} = - \frac{1669}{2}$$

a17 = -802

$$a_{17} = -802$$

a18 = -1539/2

$$a_{18} = - \frac{1539}{2}$$

a19 = -737

$$a_{19} = -737$$

a20 = -1409/2

$$a_{20} = - \frac{1409}{2}$$

a21 = -672

$$a_{21} = -672$$

a22 = -1279/2

$$a_{22} = - \frac{1279}{2}$$

a23 = -607

$$a_{23} = -607$$

a24 = -1149/2

$$a_{24} = - \frac{1149}{2}$$

a25 = -542

$$a_{25} = -542$$

a26 = -1019/2

$$a_{26} = - \frac{1019}{2}$$

a27 = -477

$$a_{27} = -477$$

a28 = -889/2

$$a_{28} = - \frac{889}{2}$$

a29 = -412

$$a_{29} = -412$$

a30 = -759/2

$$a_{30} = - \frac{759}{2}$$

a31 = -347

$$a_{31} = -347$$

a32 = -629/2

$$a_{32} = - \frac{629}{2}$$

a33 = -282

$$a_{33} = -282$$

a34 = -499/2

$$a_{34} = - \frac{499}{2}$$

a35 = -217

$$a_{35} = -217$$

a36 = -369/2

$$a_{36} = - \frac{369}{2}$$

a37 = -152

$$a_{37} = -152$$

a38 = -239/2

$$a_{38} = - \frac{239}{2}$$

a39 = -87

$$a_{39} = -87$$

a40 = -109/2

$$a_{40} = - \frac{109}{2}$$

a41 = -22

$$a_{41} = -22$$

a42 = 21/2

$$a_{42} = \frac{21}{2}$$

a43 = 43

$$a_{43} = 43$$

...

Сумма

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

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

S43 = -54997/2

$$S_{43} = - \frac{54997}{2}$$