Tính tổng: S=10.17+11.18+12.19+...+100.107+101.108
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Đặt A =4/1x3 + 6/3x6+8/6x10 +14/10x17 +12/17x23 . => A/2 = 2/1x3 +3/3x6 +4/6x10+7/10x17 +6/17x23. =>A/2 = 1-1/3 + 1/3-1/6+1/6-1/10+1/10-1/17+1/17-1/23. => A/2=1-1/23 . => A/2=22/23 . => A=44/23 . cho mik nha
\(A=\frac{4}{1.3}+\frac{6}{3.6}+\frac{8}{6.10}+\frac{14}{10.17}+\frac{12}{17.23}\)
\(=2\left(\frac{2}{1.3}+\frac{3}{3.6}+\frac{4}{6.10}+\frac{7}{10.17}+\frac{6}{17.23}\right)\)
\(=2\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{10}+\frac{1}{10}-\frac{1}{17}+\frac{1}{17}-\frac{1}{23}\right)\)
\(=2\left(1-\frac{1}{23}\right)\)
\(=2\cdot\frac{22}{23}=\frac{44}{23}\)
a, Ta thấy :
\(\dfrac{2}{4.11}\)=\(\dfrac{2}{4\left(4+7\right)}\)
\(\dfrac{2}{11.18}\) = \(\dfrac{2}{11\left(11+7\right)}\)
\(\dfrac{2}{18.25}\) = \(\dfrac{2}{15\left(15+7\right)}\)
.................
Vậy số hạng tổng quát của dãy số trên là : \(\dfrac{2}{a\left(a+7\right)}\).
b, S= \(\dfrac{2}{4.11}\) + \(\dfrac{2}{11.18}\) +\(\dfrac{2}{18.25}\) + .........+ \(\dfrac{2}{683.690}\) + \(\dfrac{2}{697.704}\)
= \((\) \(\dfrac{2}{4}-\dfrac{2}{11}+\dfrac{2}{11}-\dfrac{2}{18}+\dfrac{2}{18}-\dfrac{2}{25}+\dfrac{2}{25}-\)......+ \(\dfrac{2}{683}-\dfrac{2}{690}+\dfrac{2}{697}-\dfrac{2}{704}\) \()\)\(\div\) 7
= \((\) \(\dfrac{2}{4}-\dfrac{2}{704}\)\()\) \(\div\) 7
=\(\dfrac{25}{352}\)
Vậy tổng của 100 số hạng đầu tiên của dãy là S=\(\dfrac{25}{352}\).
\(\dfrac{1}{7}\left(\dfrac{7}{3.10}+\dfrac{7}{10.17}+...+\dfrac{7}{73.80}-\left(\dfrac{7}{2.9}+\dfrac{7}{9.16}+...+\dfrac{7}{23.30}\right)\right)\)
\(=\dfrac{1}{7}\left(\dfrac{1}{3}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{17}+...+\dfrac{1}{73}-\dfrac{1}{80}-\left(\dfrac{1}{2}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{16}+...+\dfrac{1}{23}-\dfrac{1}{30}\right)\right)\)
\(=\dfrac{1}{7}\left(\dfrac{1}{3}-\dfrac{1}{80}-\left(\dfrac{1}{2}-\dfrac{1}{30}\right)\right)\)
\(=\dfrac{1}{7}\left(\dfrac{77}{240}-\dfrac{7}{15}\right)=\dfrac{1}{7}.\left(-\dfrac{7}{48}\right)=-\dfrac{1}{48}\)