So sánh:
a)\(\frac{7^{15}}{1+7+7^2+...+7^{14}}\) và \(\frac{9^{15}}{1+9+9^2+...+9^{14}}\)
b) \(\frac{1+3+3^2+...+3^{10}}{1+3+3^2+...+3^9}\)và \(\frac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}\)
So sánh:
a)\(\frac{7^{15}}{1+7+7^2+...+7^{14}}\) và \(\frac{9^{15}}{1+9+9^2+...+9^{14}}\)
b) \(\frac{1+3+3^2+...+3^{10}}{1+3+3^2+...+3^9}\)và \(\frac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}\)
b, Ta có:\(\dfrac{1+3+3^2+.....+3^{10}}{1+3+3^2+.....+3^9}\) \(=\dfrac{1}{1+3+3^2+...+3^9}+\dfrac{3+3^2+...+3^{10}}{1+3+3^2+...+3^9}\)\(=\dfrac{1}{1+3+3^2+...+3^9}+\dfrac{3.\left(1+3+3^2+...+3^9\right)}{1+3+3^2+...+3^9}\)
\(=\dfrac{1}{1+3+3^2+...+3^9}+3< 4\)
\(\Rightarrow\) \(\dfrac{1+3+3^2+...+3^{10}}{1+3+3^2+...+3^9}< 4\) \(\left(1\right)\)
Ta có :\(\dfrac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}\)
\(=\dfrac{1}{1+5+5^2+...+5^9}+\dfrac{5+5^2+...+5^{10}}{1+5+5^2+....+5^9}\)
\(=\dfrac{1}{1+5+5^2+...+5^9}+\dfrac{5.\left(1+5+5^2+...+5^9\right)}{1+5+5^2+...+5^9}\)
\(=\dfrac{1}{1+5+5^2+...+5^9}+5>5\)
\(\Rightarrow\) \(\dfrac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}>5\) \(\left(2\right)\)
Từ \(\left(1\right)và\left(2\right)\)
\(\Rightarrow\dfrac{1+3+3^2+...+3^{10}}{1+3+3^2+...+3^9}< \dfrac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}\)
Vậy \(\dfrac{1+3+3^2+...+3^{10}}{1+3+3^2+...+3^9}< \dfrac{1+5+5^2+...+5^{10}}{1+5+5^2+...+5^9}\)
a, Đặt \(A\)\(=\dfrac{7^{15}}{1+7+7^2+...+7^{14}}\)
\(\Rightarrow\) \(\dfrac{1}{A}\) \(=\dfrac{1+7+7^2+...+7^{14}}{7^{15}}=\dfrac{1}{7^{15}}+\dfrac{7}{7^{15}}+\dfrac{7^2}{7^{15}}+...+\dfrac{7^{14}}{7^{15}}\)
\(=\dfrac{1}{7^{15}}+\dfrac{1}{7^{14}}+\dfrac{1}{7^{13}}+....+\dfrac{1}{7}\)
Đặt \(B=\dfrac{9^{15}}{1+9+9^2+...+9^{14}}\)
\(\Rightarrow\dfrac{1}{B}=\dfrac{1+9+9^2+...+9^{14}}{9^{15}}=\dfrac{1}{9^{15}}+\dfrac{9}{9^{15}}+\dfrac{9^2}{9^{15}}+...+\dfrac{9^{14}}{9^{15}}\)
\(=\dfrac{1}{9^{15}}+\dfrac{1}{9^{14}}+\dfrac{1}{9^{13}}+...+\dfrac{1}{9}\)
Mà \(\dfrac{1}{7^{15}}>\dfrac{1}{9^{15}};\dfrac{1}{7^{14}}>\dfrac{1}{9^{14}};\dfrac{1}{7^{13}}>\dfrac{1}{9^{13}};....;\dfrac{1}{7}>\dfrac{1}{9}\)
\(\Rightarrow\dfrac{1}{A}>\dfrac{1}{B}\) \(\Rightarrow A< B\)
Vậy\(\dfrac{7^{15}}{1+7+7^2+...+7^{14}}>\dfrac{9^{15}}{1+9+9^2+....+9^{14}}\)
Mình sửa kết luận
Vậy\(\dfrac{7^{15}}{1+7+7^2+...+7^{14}}< \dfrac{9^{15}}{1+9+9^2+...+9^{14}}\)
So sánh 1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+...+1/13*14 và 1
\(A=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{13.14}\)
\(=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{13}-\dfrac{1}{14}\)
\(=\dfrac{1}{1}-\dfrac{1}{14}=\dfrac{13}{14}\)
Ta thấy: \(\dfrac{13}{14}< 1\)
Vậy A<1
Đặt \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+...+\dfrac{1}{13\cdot14}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{13}-\dfrac{1}{14}\)
\(=1-\dfrac{1}{14}\)
\(=\dfrac{14}{14}-\dfrac{1}{14}\)
\(=\dfrac{13}{14}< 1\)
Vậy A < 1
Ta có :B= \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{13.14}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+....+\dfrac{1}{13}-\dfrac{1}{14}\)
\(=1-\dfrac{1}{14}\)
Vì \(1-\dfrac{1}{14}< 1\)
Suy ra : \(B< 1\)
cho A= 1/4 + 1/5 + 1/6 + ... + 1/63. so sánh A và 2
A= (1/4+1/5+...+1/9)+(1/10+...+1/19)+(1/20+...+1/29)+(1/30+...+1/39)+....+(1/50+1/63)
a>1/9.6+ 1/20.10+1/30.10+... + 1/60.10
A>2/3+1/2+1/3+1/4+1/5+1/6
A>1+1/2+1/4+1/5+1/6
A>1+67/60
=> A>2+7/60
=> A>2
A=2000/2001+2001/2002, B=2000+2001/2001+2002
So sánh A và B
B= \(\dfrac{2000+2001}{2001+2002}=\dfrac{2000}{2001+2002}+\dfrac{2001}{2001+2002}\)
Ta có : \(\dfrac{2000}{2001}>\dfrac{2000}{2001+2002};\dfrac{2001}{2002}>\dfrac{2001}{2001+2002}\)
\(\Rightarrow\) \(\dfrac{2000}{2001}+\dfrac{2001}{2000}>\dfrac{2000+2001}{2001+2002}\)
\(\Rightarrow\) A>B
Cho A =\(\dfrac{1}{1.2}+\dfrac{1}{3.4}+\dfrac{1}{5.6}+...+\dfrac{1}{217.218}\) và B=\(\dfrac{1}{110}+\dfrac{1}{111}+\dfrac{1}{112}+...+\dfrac{1}{218}\)
So sánh A và B.
A = \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{217.218}\)
A = \(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{217}-\dfrac{1}{218}\)
A = 1 - \(\dfrac{1}{218}\)
B = \(\dfrac{1}{110}\) + \(\dfrac{1}{111}\) + \(\dfrac{1}{112}\) + ... + \(\dfrac{1}{218}\)
Xét dãy số 110; 111; 112; ...; 218, dãy số này có số số hạng là:
(218 - 110) : 1 + 1 = 109 (số)
Mặt khác \(\dfrac{1}{110}\) > \(\dfrac{1}{111}>\dfrac{1}{112}>...>\dfrac{1}{218}\)
⇒ B = \(\dfrac{1}{110}\) + \(\dfrac{1}{111}\) + \(\dfrac{1}{112}+...+\dfrac{1}{218}\) < \(\dfrac{1}{110}\) + \(\dfrac{1}{110}\)+ ... +\(\dfrac{1}{110}\)
B < \(\dfrac{1}{110}\) x 109
B < 1 - \(\dfrac{1}{110}\)
\(\dfrac{1}{128}\) < \(\dfrac{1}{110}\) ⇒ A = 1 - \(\dfrac{1}{128}\) > 1 - \(\dfrac{1}{110}\) > B
A > B
So sánh 10^n/ 10^n+1 và 10^n+1/ 10^n+2
Giải:
Ta có:
\(\dfrac{10^n}{10^n+1}=\dfrac{10^n+1-1}{10^n+1}=1-\dfrac{1}{10^n+1}\)
\(\dfrac{10^n+1}{10^n+2}=\dfrac{10^n+2-1}{10^n+2}=1-\dfrac{1}{10^n+2}\)
Vì \(\dfrac{1}{10^n+1}>\dfrac{1}{10^n+2}\)
\(\Leftrightarrow-\dfrac{1}{10^n+1}< -\dfrac{1}{10^n+2}\)
\(\Leftrightarrow-\dfrac{1}{10^n+1}+1< -\dfrac{1}{10^n+2}+1\)
Hay \(1-\dfrac{1}{10^n+1}< 1-\dfrac{1}{10^n+2}\)
\(\Leftrightarrow\dfrac{10^n}{10^n+1}< \dfrac{10^n+1}{10^n+2}\)
Vậy ...
Chúng minh:
S= \(\dfrac{1}{5}+\dfrac{1}{3}+\dfrac{1}{14}+\dfrac{1}{15}+\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}< \dfrac{1}{2}\)
Sai đề rồi \(\dfrac{1}{13}\) chứ đâu phải \(\dfrac{1}{3}\)
Ta có: \(S=\dfrac{1}{5}+\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}+\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\)
\(S=\dfrac{1}{5}+\left(\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+\dfrac{1}{63}\right)\)
\(< \dfrac{1}{5}+\left(\dfrac{1}{13}+\dfrac{1}{13}+\dfrac{1}{13}\right)+\left(\dfrac{1}{61}+\dfrac{1}{61}+\dfrac{1}{61}\right)\)
\(=\dfrac{1}{5}+\dfrac{1}{13}.3+\dfrac{1}{61}.3\)
\(=\dfrac{1}{5}+\dfrac{3}{13}+\dfrac{3}{61}\)
\(< \dfrac{1}{5}+\dfrac{3}{12}+\dfrac{3}{60}\)
\(=\dfrac{1}{5}+\dfrac{1}{4}+\dfrac{1}{20}\)
\(=\dfrac{10}{20}\)\(=\dfrac{1}{2}\)
Vậy S\(< \dfrac{1}{2}\) (đpcm)
Cho A=\(\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{200}\)
CMR: a) A>\(\dfrac{7}{12}\)
b) A>\(\dfrac{5}{8}\)
a. ta có
1/101 > 1/150
1/102> 1/150
...>1/150
1/150 = 1/150
=> 1/101 + 1/102 + .... + 1/150 > 1/150 +1/150+....+1/150(50 số hạng )= 1/3
ta có
1/151 >1/200
1/152 > 1/200
..>1/200
1/200 = 1/200
=> 1/151 + 1/152+....+1/200 > 1/200+1/200+ ...+1/200( 50 số hạng) = 1/4
==> 1/101 + 1/102+....+1/200 > 1/3 +1/4
==> A > 7/12
b, A =(1/101+1/102+....+1/150)+(1/151+1/152+.....+1/200)
A>1/150.50+1/200.50=1/3+1/4=7/12
b tách A thành bốn nhóm rồi cũng làm như trên,ta có
A>25/125+25/150+25/175+25/200=(1/5+1/6+1/7)+1/8
=107/210+1/8>1/2+1/8=5/8
So sánh các phân số sau:
a) 3/4 và 13/16
b) 4/-5 và -17/20
c) -15/16 và 1/4
d) 10/11 và 11/12
bài này ta quy đồng mẫu rồi so sánh là xong
So sánh : \(\dfrac{n+1}{n+2}\)và \(\dfrac{n}{n+3}\)