Cho N= 1/5+1/5^2+1/5^3+1/5^4+...+1/5^99.
Chúng minh N<1/14
a) thu gọn biểu thức sau: a= 5 - 5^2 + 5^3 - 5^4 +...- 5^98 + %^99
b) chứng minh rằng với mọi n thuộc N thì (2^n+1).(2^n+2) đều chia hết cho 3
c) chúng minh: A= 1/1^2 + 1/2^2+ 1/3^2+.....+1/99^2+ 1/100^2 < 1 3/4 (hỗn số)
cho N=1/5+1/5^2+1/5^3+1/5^4+...+1/5^99. chung minh N<1/4
B) B= 2!/3! + 2!/4! +...+2!/n! < 1
Bài 3 Cho C = 1/41 + 1/42 + .... + 1/80 Chứng minh 7/12 < C < 5/6
Bài 4 Tìm n thuộc số nguyên biết :
A = 19/n-1 nhân n/9 sao cho thuộc số nguyên
Bài 5 Tính A) 1/3 + 1/3^2 + 1/3^2 + .... + 1/3^100
B) 1/5 - 1/5^2 + 1/5^3 - 1/5^4 + ..... + 1/5^99 - 1/5^100
1, Thực hiện phép tính bằng cách hợp lý:
A=(1)/(2)-(2)/(5)+(1)/(3)+(5)/(7)-(-1)/(6)+(-4)/(35)+(1)/(41)
2, Chứng minh rằng:
a, 1+4+4^2+4^3+...+4^99 chia hết cho 5
b, 3^n+2-2^n+2+3^n-2^n chia hết cho 10 (với n thuộc N*)
1. \(A=\frac{1}{2}-\frac{2}{5}+\frac{1}{3}+\frac{5}{7}-\frac{-1}{6}+\frac{-4}{35}+\frac{1}{41}\)
\(=\frac{1}{2}-\frac{2}{5}+\frac{1}{3}+\frac{5}{7}+\frac{1}{6}-\frac{4}{35}+\frac{1}{41}\)
\(=\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}\right)-\left(\frac{2}{5}-\frac{5}{7}+\frac{4}{35}\right)+\frac{1}{41}\)
\(=\left(\frac{5}{6}+\frac{1}{6}\right)-\left(\frac{-11}{35}+\frac{4}{35}\right)+\frac{1}{41}\)\(=1-\frac{-7}{35}+\frac{1}{41}=1+\frac{1}{5}+\frac{1}{41}=\frac{251}{205}\)
2. a) \(1+4+4^2+4^3+......+4^{99}=\left(1+4\right)+\left(4^2+4^3\right)+.......+\left(4^{98}+4^{99}\right)\)
\(=\left(1+4\right)+4^2\left(1+4\right)+.........+4^{98}\left(1+4\right)\)
\(=5+4^2.5+........+4^{98}.5=5\left(1+4^2+.....+4^{98}\right)⋮5\)( đpcm )
b) \(3^{n+2}-2^{n+2}+3^n-2^n=\left(3^{n+2}+3^n\right)-\left(2^{n+2}+2^n\right)\)
\(=3^n\left(3^2+1\right)-2^n\left(2^2+1\right)=3^n\left(9+1\right)-2^n\left(4+1\right)\)
\(=3^n.10-2^n.5=3^n.10-2^{n-1+1}.5=3^n.10-2^{n-1}.2.5\)
\(=3^n.10-2^{n-1}.10=10\left(3^n-2^{n-1}\right)⋮10\)( đpcm )
cho M = 1/3 + 1/5 + 1/9 + 1/17 + 1/33 + 1/65 va N = 1/5^2 - 2/5^3 + 3/5^4 - 4/5^5 + . . . + 99/5^100 - 100/5^101
Tính
A=1/2+1/2^2+1/2^3+...+1/2^100
Tính
B=1/2+1/2^2+1/2^3+1/2^4+...+1/2^99 - 1/2^100
Tính
C=1/2+1/2^3+1/2^5+...+1/2^99
Tính
D=2/3+8/9+26/27+...+3^n-1/3^n.Chứng minh A>n-1/2
Tính: E=4/3+10/9+28/27+...+3^39+1/3^92.Chứng minh B<100
Tính
F=5/4+5/4^2+5/4^3+...+5/4^99.Chứng minh C<5/3
Tính
G=3/1^2*2^2+5/2^2*3^2+7/3^2*4^2+...+19/9^2*10^2.Chứng Minh D<1
a) Ta có: \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\)
\(\Leftrightarrow2\cdot A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}\)
\(\Leftrightarrow2\cdot A-A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\right)\)
\(\Leftrightarrow A=1-\frac{1}{2^{100}}\)
Bài 1 Tìm x,y thuộc số nguyên
5/x - y/3 =1/6
Bài 2 Chứng minh
A) A= 1/4^2 + 1/6^2 + .... + 1/(2^n)^2 < 1/4
B) B= 2!/3! + 2!/4! +...+2!/n! < 1
Bài 3 Cho
C = 1/41 + 1/42 + .... + 1/80
Chứng minh 7/12 < C < 5/6
Bài 4 Tìm n thuộc số nguyên biết :
A = 19/n-1 nhân n/9 sao cho thuộc số nguyên
Bài 5 Tính
A) 1/3 + 1/3^2 + 1/3^2 + .... + 1/3^100
B) 1/5 - 1/5^2 + 1/5^3 - 1/5^4 + ..... + 1/5^99 - 1/5^100
Bài 1:
\(\dfrac{5}{x} - \dfrac{y}{3} =\dfrac{1}{6}\)
\(\Rightarrow\dfrac{1}{6}+\dfrac{y}{3}=\dfrac{5}{x}\)
\(\Rightarrow\dfrac{1}{6}+\dfrac{2y}{6}=\dfrac{5}{x}\)
\(\Rightarrow1+\dfrac{2y}{6}=\dfrac{5}{x}\)
\(\Rightarrow x.\left(1+2y\right)=30\)
Vì \(2y\) chẵn nên \(1+2y\) lẻ
\(\Rightarrow1+2y\in\left\{\pm1;\pm3;\pm5;\pm30\right\}\)
\(\Rightarrow x\in\left\{\pm10;\pm30;\pm6;\pm2\right\}\)
Bài 2:
\(\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{2.4}+\dfrac{1}{4.6}+\dfrac{1}{6.8}+...+\dfrac{1}{\left(2n-2\right).2n}\)
\(=\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{\left(2n-2\right).2n}\right).\dfrac{1}{2}\)
\(=\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{12}+...+\dfrac{1}{2n-2}-\dfrac{1}{2n}\right).\dfrac{1}{2}\)
\(=\left(\dfrac{1}{2}-\dfrac{1}{2n}\right).\dfrac{1}{2}\)
\(=\dfrac{1}{4}-\dfrac{1}{2n.2}< \dfrac{1}{4}\)
\(\Rightarrow\dfrac{1}{4^2}+\dfrac{1}{6^2}+\dfrac{1}{8^2}+...+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{4}\left(đpcm\right)\)
Câu B dấu chấm than là kí hiệu gì thế bạn?
Tinh nhanh:
m)5^1+5^2+5^3+...+5^199+5^200
n)3^0-3^2+3^3-3^4+...+3^2017-3^2018+3^2019-3^2020
o)6+6*9+6*9^2+6*9^3+...+6*9^99
p)(-1)*(-1^2)*(-1^3)*(-1^4)*...*(-1^99)*-(1^100)
Giup minh nhe!
Đặt \(A=5+5^2+5^3+....+5^{199}+5^{200}\)
\(\Leftrightarrow5A=5\left(5+5^2+5^3+....+5^{199}+5^{200}\right)\)
\(\Leftrightarrow5A=5^2+5^3+5^4+....+5^{200}+5^{201}\)
\(\Leftrightarrow5A-A=\left(5^2+5^3+5^4+....+5^{200}+5^{201}\right)-\left(5+5^2+5^3+....+5^{199}+5^{200}\right)\)
\(\Leftrightarrow4A=5^{201}-5\)
\(\Leftrightarrow A=\frac{5^{201}-5}{4}\)
Cho S = 1+5+5^2+5^3+...+5^99+5^100
1.S có chia hết cho 3 không ? Vì sao ?
2.Tìm số tự nhiên n biết 4*S+1=5n+1