Ejercicio 288

octubre 30, 2019abril 16, 2019 por Gestor Baldor

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CAPITULO XXXVII

Progresiones

Ejercicio 288

Hallar la suma de los:

8 primeros términos de $\div 15.19.23…..$
$\begin{matrix} S e a & : \\ a = 15, n = 8, r = 19 - 15 = 4 \\ u & = a + (n - 1) r \\ u & = 15 + (8 - 1) 4 \\ u & = 15 + 7 (4) \\ u & = 15 + 28 \\ u & = 43 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(15 + 43)}{2} \\ S & = (58) 4 \\ S & = 232 \end{matrix}$
19 primeros términos de $\div 31.38.45…..$
$\begin{matrix} S e a & : \\ a = 31, n = 19, r = 38 - 31 = 7 \\ u & = a + (n - 1) r \\ u & = 31 + (19 - 1) 7 \\ u & = 31 + 18 (7) \\ u & = 31 + 126 \\ u & = 157 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(31 + 157) 19}{2} \\ S & = \frac{19}{2} \\ S & = 1786 \end{matrix}$
24 primeros términos de $\div 42.32.22…..$
$\begin{matrix} S e a & : \\ a = 42, n = 24, r = 32 - 42 = - 10 \\ u & = a + (n - 1) r \\ u & = 42 + (24 - 1) (- 10) \\ u & = 42 + (23) (- 10) \\ u & = 42 - 230 \\ u & = - 188 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(42 - 188)}{2} \\ S & = (- 146) 12 \\ S & = - 1752 \end{matrix}$
80 primeros términos de $\div - 10. - 6. - 2…..$
$\begin{matrix} S e a & : \\ a = - 10, n = 80, r = - 6 + 10 = 4 \\ u & = a + (n - 1) r \\ u & = - 10 + (80 - 1) 4 \\ u & = - 10 + (79) 4 \\ u & = - 10 + 316 \\ u & = 306 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(- 10 + 306)}{2} \\ S & = (296) 40 \\ S & = 11840 \end{matrix}$
60 primeros términos de $\div 11.1. - 9…..$
$\begin{matrix} S e a & : \\ a = 11, n = 60, r = 1 - 11 = - 10 \\ u & = a + (n - 1) r \\ u & = 11 + (60 - 1) (- 10) \\ u & = 11 + (59) (- 10) \\ u & = 11 - 590 \\ u & = - 579 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(11 - 579)}{2} \\ S & = (- 568) 30 \\ S & = - 17040 \end{matrix}$
50 primeros términos de $\div - 5. - 13. - 21…..$
$\begin{matrix} S e a & : \\ a = - 5, n = 50, r = - 13 + 5 = - 8 \\ u & = a + (n - 1) r \\ u & = - 5 + (50 - 1) (- 8) \\ u & = - 5 + (49) (- 8) \\ u & = - 5 - 392 \\ u & = - 397 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(- 5 - 397)}{2} \\ S & = (- 402) 25 \\ S & = - 10050 \end{matrix}$
9 primeros términos de $\div \frac{1}{2} .1. \frac{3}{2} …..$
$\begin{matrix} S e a & : \\ a = \frac{1}{2}, n = 9, r = 1 - \frac{1}{2} = \frac{1}{2} \\ u & = a + (n - 1) r \\ u & = \frac{1}{2} + (9 - 1) (\frac{1}{2}) \\ u & = \frac{1}{2} + () (\frac{1}{2}) \\ u & = \frac{1}{2} + 4 \\ u & = \frac{9}{2} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{1}{2} + \frac{9}{2}) 9}{2} \\ S & = \frac{(\frac{}{2}) 9}{2} \\ S & = \frac{45}{2} \leftrightarrow 22 \frac{1}{2} \end{matrix}$
14 primeros términos de $\div \frac{3}{10} . \frac{2}{5} . \frac{1}{2} …..$
$\begin{matrix} S e a & : \\ a = \frac{3}{10}, n = 14, r = \frac{2}{5} - \frac{3}{10} = \frac{1}{10} \\ u & = a + (n - 1) r \\ u & = \frac{3}{10} + (14 - 1) (\frac{1}{10}) \\ u & = \frac{3}{10} + (13) (\frac{1}{10}) \\ u & = \frac{3}{10} + \frac{13}{10} \\ u & = \frac{}{} \\ u & = \frac{8}{5} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{3}{10} + \frac{8}{5})}{2} \\ S & = (\frac{19}{10}) 7 \\ S & = \frac{133}{10} \leftrightarrow 13 \frac{3}{10} \end{matrix}$
19 primeros términos de $\div \frac{3}{4} . \frac{3}{2} . \frac{9}{4} …..$
$\begin{matrix} S e a & : \\ a = \frac{3}{4}, n = 19, r = \frac{3}{2} - \frac{3}{4} = \frac{3}{4} \\ u & = a + (n - 1) r \\ u & = \frac{3}{4} + (19 - 1) (\frac{3}{4}) \\ u & = \frac{3}{4} + () (\frac{3}{}) \\ u & = \frac{3}{4} + \frac{27}{2} \\ u & = \frac{57}{4} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{3}{4} + \frac{57}{4}) 19}{2} \\ S & = \frac{(\frac{}{4}) 19}{2} \\ S & = \frac{15 (19)}{2} \\ S & = \frac{285}{2} \leftrightarrow 142 \frac{1}{2} \end{matrix}$
34 primeros términos de $\div \frac{2}{5} . \frac{7}{55} …..$
$\begin{matrix} S e a & : \\ a = \frac{2}{5}, n = 34, r = \frac{7}{55} - \frac{2}{5} = - \frac{3}{11} \\ u & = a + (n - 1) r \\ u & = \frac{2}{5} + (34 - 1) (- \frac{3}{11}) \\ u & = \frac{2}{5} + () (- \frac{3}{11}) \\ u & = \frac{2}{5} - 9 \\ u & = - \frac{43}{5} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{2}{5} - \frac{43}{5})}{2} \\ S & = (- \frac{41}{5}) 17 \\ S & = - \frac{697}{5} \leftrightarrow - 139 \frac{2}{5} \end{matrix}$
11 primeros términos de $\div 2 \frac{1}{3} .3 \frac{2}{15} …..$
$\begin{matrix} S e a & : \\ a = \frac{7}{3}, n = 11, r = \frac{47}{15} - \frac{7}{3} = \frac{4}{5} \\ u & = a + (n - 1) r \\ u & = \frac{7}{3} + (11 - 1) (\frac{4}{5}) \\ u & = \frac{7}{3} + () (\frac{4}{5}) \\ u & = \frac{7}{3} + 8 \\ u & = \frac{31}{3} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{7}{3} + \frac{31}{3}) 11}{2} \\ S & = \frac{(\frac{}{3}) 11}{2} \\ S & = (\frac{19}{3}) 11 \\ S & = \frac{209}{3} \leftrightarrow 69 \frac{2}{3} \end{matrix}$
46 primeros términos de $\div 3 \frac{1}{4} .3 \frac{13}{20} ……$
$\begin{matrix} S e a & : \\ a = \frac{13}{4}, n = 46, r = \frac{73}{20} - \frac{13}{4} = \frac{2}{5} \\ u & = a + (n - 1) r \\ u & = \frac{13}{4} + (46 - 1) (\frac{2}{5}) \\ u & = \frac{13}{4} + () (\frac{2}{5}) \\ u & = \frac{13}{4} + 18 \\ u & = \frac{85}{4} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(\frac{13}{4} + \frac{85}{4})}{3} \\ S & = (\frac{}{}) 23 \\ S & = \frac{1127}{2} \leftrightarrow 563 \frac{1}{2} \end{matrix}$
17 primeros términos de $\div - 2. \frac{1}{4} ……$
$\begin{matrix} S e a & : \\ a = - 2, n = 17, r = \frac{1}{4} + 2 = \frac{9}{4} \\ u & = a + (n - 1) r \\ u & = - 2 + (17 - 1) (\frac{9}{4}) \\ u & = - 2 + () (\frac{9}{4}) \\ u & = - 2 + 36 \\ u & = 34 \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(- 2 + 34) 17}{2} \\ S & = \frac{(17)}{2} \\ S & = 272 \end{matrix}$
12 primeros términos de $\div - 5. - 4 \frac{5}{8} ……$
$\begin{matrix} S e a & : \\ a = - 5, n = 12, r = - \frac{37}{8} + 5 = \frac{3}{8} \\ u & = a + (n - 1) r \\ u & = - 5 + (12 - 1) (\frac{3}{8}) \\ u & = - 5 + (11) (\frac{3}{8}) \\ u & = - 5 + \frac{33}{8} \\ u & = - \frac{7}{8} \\ S & = \frac{(a + u) n}{2} \\ S & = \frac{(- 5 - \frac{7}{8})}{2} \\ S & = (- \frac{47}{}) \\ S & = - \frac{141}{4} \leftrightarrow - 35 \frac{1}{4} \end{matrix}$