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2:
a: A=1+2+2^2+2^3+2^4
=>2A=2+2^2+2^3+2^4+2^5
=>A=2^5-1
=>A=B
b: C=3+3^2+...+3^100
=>3C=3^2+3^3+...+3^101
=>2C=3^101-3
=>\(C=\dfrac{3^{101}-3}{2}\)
=>C=D
Ta có:
\(\left\{\begin{matrix}5^{27}=\left(5^3\right)^9=125^9\\2^{63}=\left(2^7\right)^9=128^9\end{matrix}\right\}\Rightarrow5^{27}< 2^{63}\left(1\right)\)
\(\left\{\begin{matrix}2^{63}=\left(2^9\right)^7=512^7\\5^{28}=\left(5^4\right)^7=625^7\end{matrix}\right\}\Rightarrow2^{63}< 5^{28}\left(2\right)\)
Từ (1) và (2) \(\Rightarrow5^{27}< 2^{63}< 5^{28}\) (đpcm)
1.
Ta có:
1/2 < 2/3
3/4 < 4/5
.............
99/100 < 100/101
=> 1/2*3/4*5/6*...*99/100 < 2/3*4/5*6/7*...*100/101
=> A < B
2.
\(A\cdot B=\left[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot...\cdot\frac{99}{100}\right]\cdot\left[\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot...\cdot\frac{100}{101}\right]\)
\(A\cdot B=\frac{\left[1\cdot3\cdot5\cdot7\cdot...\cdot99\right]\left[2\cdot4\cdot6\cdot8\cdot...\cdot100\right]}{\left[2\cdot4\cdot6\cdot8\cdot...\cdot100\right]\left[3\cdot5\cdot7\cdot9\cdot...\cdot101\right]}=\frac{1\cdot3\cdot5\cdot...\cdot99}{3\cdot5\cdot7\cdot...\cdot101}=\frac{1}{101}\)
3.
Vì A < B => A.A < A.B => A2 < 1/101 < 1/100
Mà A2 < 1/100 <=> A2 < \(\frac{1}{10}^2\)=> A < 1/10
M=1+ 3^100/1+3+3^2+..+3^99
=1+1: 1+3+3^2+...+3^99/3^100
=1+1:(1/3^100+1/3^99+..+1/3)
tương tự ta có
N=1+1: (1/5^100+1/5^99+......+1/5)
do 1/5^100<1/3^100;1/5^99<1/3^99,...,1/5<1/3
=M<N
\(A=1+5+5^2+5^3+...+5^{59}\)
\(=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+...+\left(5^{57}+5^{58}+5^{59}\right)\)
\(=\left(1+5+5^2\right)+5^3\left(1+5+5^2\right)+...+5^{57}\left(1+5+5^2\right)\)
\(=31\left(1+5^3+...+5^{57}\right)\)chia hết cho \(31\).
\(A=1+5+5^2+5^3+...+5^{59}\)
\(5A=5+5^2+5^3+5^4+...+5^{60}\)
\(5A-A=\left(5+5^2+5^3+5^4+...+5^{60}\right)-\left(1+5+5^2+5^3+...+5^{59}\right)\)
\(4A=5^{60}-1\)
\(A=\frac{5^{60}-1}{4}< \frac{5^{60}}{4}\).
Bạn ơi câu b bạn vt thiếu đề r
Chứng tỏ j v ??
a, \(3^{210}\) và \(2^{350}\)
Ta có \(\hept{\begin{cases}3^{210}=\left(3^3\right)^{70}=27^{70}\\2^{350}=\left(2^5\right)^{70}=32^{70}\end{cases}}\)
Mà 32 > 27 > 0
\(\Rightarrow32^{70}>27^{70}\)
\(\Rightarrow2^{350}>3^{210}\)
Vậy \(3^{210}< 2^{350}\)
b, Thiếu đề ròi
~~~~~ Học tốt ~~~~~~~