Cho G = 5/3 + 8/32 + 11/33 + ........ + 302/3100 . Chứng minh rằng 2 5/9 ( hỗn số) < G < 3 1/2 ( hỗn số )
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2
T
5 tháng 3 2022
3G = 3(5/3+8/3^2+11/3^3+...+302/3^100) = 5 + 8/3 + 11/3^2 + ... + 302/3^99
3G - G = ( 5 + 8/3 + 11/3^2 + ... + 302/3^99 ) - ( 5/3+8/3^2+11/3^3+...+302/3^100 )
2G = 5 + 8/3 + 11/3^2 + ... + 302/3^99 - 5/3 - 8/3^2 - 11/3^3 - ... - 302/3^100
2G = 5 + 1 + 1/3 + 1/3^2 + ... + 1/3^98 - 302/3^100 (1)
Đặt B = 1/3 + 1/3^2 + 1/3^3 + ... + 1/3^98
3B = 1 + 1/3 + 1/3^2 + ... + 1/3^97
3B - B = 1 + 1/3 + 1/3^2 + ... + 1/3^97 - 1/3 - 1/3^2 - 1/3^3 - ... - 1/3^98
2B = 1 - 1/3^98
B = 1/2 - 1/3^98 (2)
Từ (1) và (2) => 2G = 5 + 1 + 1/3 + 1/3^2 + ... + 1/3^98 - 302/3^100 = 6 + 1/2 - 1/3^98
=> G = 3 + 1/4 - 1/3^98
ta có 0 < 1/4 - 1/3^98 < 1/2
=> 3 < 3 + 1/4 - 1/3^98 < 3 + 1/2
suy ra 2 + 5/9 < G < 3 + 1/2
Hôm kia
bạn có: 3G = (5 + 8/3) + (11/3^2 + 14/3^3 + ... + 299/3^98 + 302/3^99)
G = 5/3 + (8/3^2 + 11/3^3 + .... + 296/3^98 + 299/3^99 + 302/3^100)
bạn có 3G - G = 5 + 8/3 - 5/3 + (11/3^2 - 8/3^2) + (14/3^3 - 11/3^3) + .... + (299/3^98 - 296/3^98) + (302/3^99 - 299/3^99) - 302/3^100
hay 2G = 5 +8/3 - 5/3 + (3/3^2 + 3/3^3 + ... + 3/3^98 + 3/3^99) - 302/3^100
2G = 6 + (1/3 + 1/3^2 +... + 1/3^97 + 1/3^98)
đặt H = 1/3 + 1/3^2 + ... + 1/3^97 + 1/3^98
suy ra ta có 3H = 1 + 1/3 + .... + 1/3^96 + 1/3^97
3H - H = 1 - 1/3^98 hay 2H = 1 - 1/3^98
ở trên bạn có:
2G = 6 + (1/3 + 1/3^2 +... + 1/3^97 + 1/3^98)
hay 2G = 6 + H
hay 4G = 12 + 2H
hay 4G = 12 + 1 - 1/3^98
hay G = 13/4 - (1/3^98)/4
H9
HT.Phong (9A5)
CTVHS
22 tháng 6 2023
\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{99}}\)
\(\Rightarrow\dfrac{A}{3}=\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\)
\(\Rightarrow A-\dfrac{A}{3}=\dfrac{2A}{3}=\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)-\left(\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\dfrac{2A}{3}=\left(\dfrac{1}{3^2}-\dfrac{1}{3^2}\right)+\left(\dfrac{1}{3^3}-\dfrac{1}{3^3}\right)+...+\left(\dfrac{1}{3^{99}}-\dfrac{1}{3^{99}}\right)+\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)=\dfrac{1}{3}-\dfrac{1}{3^{100}}\)
\(\Rightarrow2A=3\cdot\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\text{A}=\dfrac{1-\dfrac{1}{3^{99}}}{2}\)
\(\Rightarrow A=\dfrac{1}{2}-\dfrac{1}{2.3^{99}}< \dfrac{1}{2}\)
6 tháng 8 2022
a: \(\Leftrightarrow\left(\dfrac{13}{4}:x\right)\cdot\left(-\dfrac{5}{4}\right)=\dfrac{-10}{6}-\dfrac{5}{6}=\dfrac{-15}{6}=\dfrac{-5}{2}\)
\(\Leftrightarrow\dfrac{13}{4}:x=\dfrac{5}{2}\cdot\dfrac{5}{4}=\dfrac{25}{8}\)
hay \(x=\dfrac{13}{4}:\dfrac{25}{8}=\dfrac{13}{4}\cdot\dfrac{8}{25}=\dfrac{26}{25}\)
b: \(\Leftrightarrow\dfrac{3}{4}:x=\dfrac{11}{36}-\dfrac{1}{4}=\dfrac{2}{36}=\dfrac{1}{18}\)
=>\(x=\dfrac{3}{4}:\dfrac{1}{18}=\dfrac{54}{4}=\dfrac{27}{2}\)
c: \(\Leftrightarrow\left(-\dfrac{6}{5}+x\right):\left(-3.6\right)=-\dfrac{7}{4}+\dfrac{1}{4}\cdot8=\dfrac{1}{4}\)
=>x-6/5=-9/10
=>x=3/10
12 tháng 8 2021
f) \(-\dfrac{1}{2}+\dfrac{11}{4}-\left(\dfrac{11}{4}-\dfrac{1}{2}\right)=-\dfrac{1}{2}+\dfrac{11}{4}-\dfrac{11}{4}+\dfrac{1}{2}=0\)
g) \(\left(\dfrac{92}{9}+\dfrac{13}{5}\right)-\dfrac{47}{9}=\left(\dfrac{92}{9}-\dfrac{47}{9}\right)+\dfrac{13}{5}=5+\dfrac{13}{5}=\dfrac{25}{5}+\dfrac{13}{5}=\dfrac{38}{5}\)
h) \(\left(\dfrac{44}{7}-\dfrac{32}{9}\right)-\left(\dfrac{37}{7}+\dfrac{4}{9}\right)=\dfrac{44}{7}-\dfrac{32}{9}-\dfrac{37}{7}-\dfrac{4}{9}=\left(\dfrac{44}{7}-\dfrac{37}{7}\right)-\left(\dfrac{32}{9}+\dfrac{4}{9}\right)=1-4=-3\)
31 tháng 10 2020
B= 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + 9 + 10 - 11 - 12 +...+ 298 - 299 - 300 + 301 + 302
= 1 + ( 2 - 3 - 4 + 5) + ( 6 - 7 - 8 + 9) + ( 10 - 11 - 12 + 13) +...+ (298 - 299 - 300 + 301 ) + 302
= 1 + 0 + 0 +...+ 0 + 302
= 1 + 302 = 303 chia hết cho 3
=> B chia hết cho 3
12 tháng 8 2021
f) -1/2 + 11/4 - (11/4 - 1/2)
= -1/2 + 11/4 - 11/4 + 1/2
= -1/2 + 0 + 1/2
= 0
g) (92/9 + 13/5) - 47/9
= 92/9 + 13/5 - 47/9
=(92/9 - 47/9) + 13/5
= 5 +13/5
= 5 và 13/5
h) (44/7 - 32/9) - (37/7 + 4/9)
= 44/7 - 32/9 - 37/7 - 4/9
= (44/7 - 37/7) + (32/9 - 4/9)
= 7 + 28
= 35
k cho mik nha
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