Question

[Ken Natasa]

Cm

a) 1/5+1/13+1/14+1/15+1/61+1/62+1/63 nhỏ hơn 1/2

b) 1/2 +1/2^2 +1/2^3 +....+1/2^20 nhỏ hơn 1 

c) 1/4+1/5+1/6+...+1/19 lớn hơn 1 

d) 3/1.4+3/4.7 +3/7.10 +...+3/40.43+3/43.46 nhỏ hơn 1

e) 1/2^2 + 1/3^2 +1/4^2 +1/5^2 +1/6^2+1/7^2+1/8^2 nhỏ hơn 1

Answer 1

[I don't need you]

b) Đặt \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\)

\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{19}}\)

\(\Rightarrow2A-A=1-\frac{1}{2^{20}}\)

\(\Rightarrow A=1-\frac{1}{2^{20}}< 1\left(đpcm\right)\)

c) ta có: \(\frac{1}{4}+\frac{1}{5}+...+\frac{1}{10}>\frac{1}{10}+\frac{1}{10}+...+\frac{1}{10}=\frac{7}{10}\) ( có 7 số 1/10)

\(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{19}>\frac{1}{19}+\frac{1}{19}+...+\frac{1}{19}=\frac{9}{19}\)  ( có 9 số 1/19)

\(\Rightarrow\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{19}>\frac{7}{10}+\frac{9}{10}=1\frac{33}{190}>1\)

=> đ p c m

Answer 2

[I don't need you]

d) \(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{40.43}+\frac{3}{43.46}\)

\(=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{40}-\frac{1}{43}+\frac{1}{43}-\frac{1}{46}\)

\(=1-\frac{1}{46}< 1\)

=> đ p c m

e) ta có: \(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};...;\frac{1}{7^2}< \frac{1}{6.7};\frac{1}{8^2}< \frac{1}{7.8}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{7^2}+\frac{1}{8^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{6.7}+\frac{1}{7.8}\)

                                                                                 \(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}\)

                                                                                 \(=1-\frac{1}{8}< 1\)

=> đ p c m

câu a mk ko bk, xl bn nhìu! :(