tính nhanh
A=\(1+3+3^2+3^3+3^4+...\)\(\) \(...+3^{100}\)
B=\(1+4^2+4^3+4^4+...\) \(...+4^{100}\)
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Bài 1:
A = 1 + 3 + 32 + ... + 3100
=> 3A = 3 + 32 + ... + 3101
=> 2A = 3101 - 1
=> A = \(\frac{3^{101}-1}{2}\)
B = 1 + 42 + 44 + ... + 4100
=> 8B = 42 + 44 + ... + 4102
=> 7B = 4102 - 1
=> B = \(\frac{4^{102}-1}{7}\)
Bài 2:
a) S1 = 22 + 42 + ... + 202
=> S1 = 22(1+22+...+102)
=> S1 = 22.385
=> S1 = 1540
b) S2 = 1002 + 2002 + ... + 10002
=> S2 = 1002(1+22+...+102)
=> S2 = 1002.385
=> S2 = 3850000
tính nhanh (2/3+3/4+5/6+...+99/100).(1/2+2/3+3/4+...+98/99)-(1/2+1/3+...+99/100).(2/3+2/4+...+98/99)
a) \(A=\frac{2}{1.4}+\frac{2}{4.7}+\frac{2}{7.10}+...+\frac{2}{31.34}\)
\(A=\frac{2}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{31}-\frac{1}{34}\right)\)
\(A=\frac{2}{3}.\left(1-\frac{1}{34}\right)\)
\(A=\frac{2}{3}\cdot\frac{33}{34}=\frac{11}{17}\)
b) \(B=\frac{3}{1}+\frac{3}{3}+\frac{3}{6}+...+\frac{3}{210}\)
\(B=\frac{6}{2}+\frac{6}{6}+\frac{6}{12}+...+\frac{6}{420}\) ( 3/1 = 6/2; 6/6=3/3;..)
\(B=\frac{6}{1.2}+\frac{6}{2.3}+\frac{6}{3.4}+...+\frac{6}{20.21}\)
\(B=6.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{20}-\frac{1}{21}\right)\)
\(B=6.\left(1-\frac{1}{21}\right)=6\cdot\frac{20}{21}=\frac{40}{7}\)
Bài 3:
a: a*S=a^2+a^3+...+a^2023
=>(a-1)*S=a^2023-a
=>\(S=\dfrac{a^{2023}-a}{a-1}\)
b: a*B=a^2-a^3+...-a^2023
=>(a+1)B=a-a^2023
=>\(B=\dfrac{a-a^{2023}}{a+1}\)
a.
(5/11 + 6/11 ) + ( 1/4 +3/4) + ( 6/25 + 9/25)
= 11/11 + 4/4 + 15/25
= 1 +1 + 3/5
= 2+3/5 = 13/5
b.
( 37/100 + 163/100) + (1/8 + 3/8 ) + 19/4 + 1/2
= 200/100 + 4/8 + ( 19/4 + 2/4)
= 2 + 1/2 + 21/4
= 31/4
+) \(A=1+3+3^2+3^3+3^4+...+3^{100}\)
\(\Leftrightarrow3A=3+3^2+3^3+3^4+3^5+....+3^{101}\)
\(\Leftrightarrow3A-A=2A=3^{101}-1\)
\(\Leftrightarrow A=\dfrac{3^{101}-1}{2}\)
\(B=1+4^2+4^3+4^4+...+4^{100}\)
\(\Leftrightarrow4B=4+4^3+4^4+..+4^{101}\)
\(\Leftrightarrow3B=4^{101}+4-4^2-1\)
\(\Leftrightarrow B=\dfrac{4^{101}-13}{3}\)