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Bài 1 Số số hạng của dãy là : (50-1):1+1=50(số hạng )
S = (50+1) x 50 : 2 = 1275
Bài 1:
$A=1.2+2.3+3.4+...+201.202$
$3A=1.2.3+2.3(4-1)+3.4(5-2)+....+201.202(203-200)$
$=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+....+201.202.203-200.201.202$
$=(1.2.3+2.3.4+3.4.5+...+201.202.203)-(1.2.3+2.3.4+....+200.201.202)$
$=201.202.203$
$\Rightarrow A=\frac{201.202.203}{3}=2747402$
Bài 2:
$S=4.5+5.6+6.7+....+100.101$
$3S=4.5(6-3)+5.6.(7-4)+6.7.(8-5)+....+100.101(102-99)$
$=4.5.6-3.4.5+5.6.7-4.5.6+6.7.8-5.6.7+....+100.101.102-99.100.101$
$=(4.5.6+5.6.7+6.7.8+...+100.101.102)-(3.4.5+4.5.6+5.6.7+...+99.100.101)$
$=100.101.102-3.4.5$
$\Rightarrow S=\frac{100.101.102-3.4.5}{3}=343380$
a: \(=\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{18\cdot19}-\dfrac{1}{19\cdot20}\)
=1/2-1/380
=179/380
b: \(=\dfrac{1}{1\cdot3}-\dfrac{1}{3\cdot5}+\dfrac{1}{3\cdot5}-\dfrac{1}{5\cdot7}+...+\dfrac{1}{21\cdot23}-\dfrac{1}{23\cdot25}\)
\(=\dfrac{1}{3}-\dfrac{1}{575}=\dfrac{572}{1725}\)
c: \(=1+\dfrac{1}{2}-\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}-\dfrac{1}{20}-\dfrac{1}{21}\)
=1-1/21
=20/21
d: \(=\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)\cdot...\cdot\left(1-\dfrac{1}{121}\right)\)
\(=\dfrac{2}{3}\cdot\dfrac{3}{4}\cdot...\cdot\dfrac{10}{11}\cdot\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot...\cdot\dfrac{12}{11}\)
\(=\dfrac{2}{11}\cdot\dfrac{12}{2}=\dfrac{12}{11}\)
1)
a)\(x+\dfrac{3}{22}=\dfrac{27}{121}.\dfrac{11}{9}\)
\(x+\dfrac{3}{22}=\dfrac{3}{11}\)
\(x=\dfrac{6}{22}-\dfrac{3}{22}\)
\(x=\dfrac{3}{22}\)
A) \(A=1+4+4^2+4^3+...+4^{26}\)
\(\Rightarrow4A=4+4^2+4^3+4^4+...+4^{27}\)
\(\Rightarrow4A-A=4^{27}-1\)
\(3A=4^{27}-1\)
\(A=\frac{4^{27}-1}{3}\)
B) \(B=3+3^3+3^5+3^7+...+3^{21}\)
\(\Rightarrow3^2B=3^3+3^5+3^7+3^9+...+3^{23}\)
\(\Rightarrow3^2B-B=3^{23}-3\)
\(8B=3^{23}-3\)
\(B=\frac{3^{23}-3}{8}\)
C) \(M=1.2+2.3+3.4+...+49.50\)
\(\Rightarrow3M=1.2.3+2.3.3+3.4.3+...+49.50.3\)
\(3M=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+49.50.\left(51-48\right)\)
\(3M=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+49.50.51-48.49.50\)
\(3M=\left(1.2.3+2.3.4+3.4.5+...+49.50.51\right)-\left(1.2.3+3.4.5+...+48.49.50\right)\)
\(3M=49.50.51\)
\(M=\left(49.50.51\right):3\)
\(M=41650\)
D) \(N=1.2.3+2.3.4+3.4.5+...+49.50.51\)
\(\Rightarrow4N=1.2.3.4+2.3.4.4+3.4.5.4+...+49.50.51.4\)
\(4N=1.2.3.\left(4-0\right)+2.3.4\left(5-1\right)+3.4.5.\left(6-2\right)+...+49.50.51.\left(52-48\right)\)
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