ta xét tổng của 1/31+...+1/40

tiếp tục 1/41+..+1/50

1/51+...+1/60

Trong 4 dãy số trên ta có 1/31> 1/32>1/33>...>1/41

=> Tổng trên < 10/31

cứ tiếp tục xét ta được S< 10/31+10/41+10/51<4/5

=> S < 4/5

Xét tương tự ta sẽ có S > 3/5

S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60)

Mà : (1/31+1/32+1/33+...+1/40) > 1/40 x 10 = 1/4 (gồm 10 số hạng) Tương tự : (1/41 + 1/42 + ...+ 1/50) > 1/5 ; (1/51 + 1/52+...+1/59+1/60) > 1/6 S > 1/4 + 1/5 + 1/6.

Trong khi đó (1/4 + 1/5 + 1/6) > 3/5 =>S > 3/5 (1) S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60) Mà : (1/31+1/32+1/33+...+1/40) < 1/31 x 10 = 10/30 = 1/3 (gồm 10 số hạng) => S < 4/5 (2)

Từ (1) và (2) => 3/5 <S<4/5

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+...+\frac{1}{60}\right)\)

Mà \(\left(\frac{1}{31}+...+\frac{1}{40}\right)>\frac{1}{40}\cdot10=\frac{1}{4}\)

Tương tự : \(\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)>\frac{1}{5}\)

\(\left(\frac{1}{51}+...+\frac{1}{60}\right)>\frac{1}{6}\)

\(S>\frac{1}{4}+\frac{1}{5}+\frac{1}{6}>\frac{3}{5}\)(*1)

Mặt khác:\(\left(\frac{1}{31}+...+\frac{1}{40}\right)< \frac{1}{31}\cdot10=\frac{1}{3}\)

\(\Rightarrow S< \frac{4}{5}\)(*2)

Từ (*1)(*2)= 3/5<S<4/5

Bài 1:

\(2^{49}=\left(2^7\right)^7=128^7;5^{21}=\left(5^3\right)^7=125^7\\ Vì:128^7>125^7\Rightarrow2^{49}>5^{21}\)

Bài 2:

\(a,S=1+3+3^2+3^3+...+3^{99}\\ =\left(1+3+3^2+3^3\right)+3^4.\left(1+3+3^2+3^3\right)+...+3^{96}.\left(1+3+3^2+3^3\right)\\ =40+3^4.40+...+3^{96}.40\\ =40.\left(1+3^4+...+3^{96}\right)⋮40\\ b,S=1+4+4^2+4^3+...+4^{62}\\ =\left(1+4+4^2\right)+4^3.\left(1+4+4^2\right)+...+4^{60}.\left(1+4+4^2\right)\\ =21+4^3.21+...+4^{60}.21\\ =21.\left(1+4^3+...+4^{60}\right)⋮21\)

Bài 1 :

\(2^{49}=\left(2^7\right)^7=128^7\)

\(5^{21}=\left(5^3\right)^7=125^7\)

mà \(125^7< 128^7\)

\(\Rightarrow2^{49}>5^{21}\)

Bài 2 :

a) \(S=1+3+3^2+3^3+...3^{99}\)

\(\Rightarrow S=\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)...+3^{96}\left(1+3+3^2+3^3\right)\)

\(\Rightarrow S=40+40.3^4+...+40.3^{96}\)

\(\Rightarrow S=40\left(1+3^4+...+3^{96}\right)⋮40\)

\(\Rightarrow dpcm\)

b) \(S=1+4+4^2+4^3+...4^{62}\)

\(\Rightarrow S=\left(1+4+4^2\right)+4^3\left(1+4+4^2\right)+...4^{60}\left(1+4+4^2\right)\)

\(\Rightarrow S=21+4^3.21+...4^{60}.21\)

\(\Rightarrow S=21\left(1+4^3+...4^{60}\right)⋮21\)

\(\Rightarrow dpcm\)

Ta có: S = \(\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+...+\frac{1}{60}\right)\)

Nhận xét: \(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}>\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}=\frac{10}{40}=\frac{1}{4}\)

                \(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}>\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}=\frac{10}{50}=\frac{1}{5}\)

                 \(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}>\frac{1}{60}+\frac{1}{60}+...+\frac{1}{60}=\frac{10}{60}=\frac{1}{6}\)

\(\Rightarrow S>\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\)

\(\Rightarrow S>\frac{37}{60}>\frac{36}{60}=\frac{3}{5}\)      (1)

Lại có: \(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}< \frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}=\frac{10}{30}=\frac{1}{3}\)

           \(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}< \frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}=\frac{10}{40}=\frac{1}{4}\)

           \(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}< \frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}=\frac{10}{50}=\frac{1}{5}\)

\(\Rightarrow S< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}\)

\(\Rightarrow S< \frac{47}{60}< \frac{48}{60}=\frac{4}{5}\)         (2)

Từ (1) và (2) => \(\frac{3}{5}< S< \frac{4}{5}\) (đpcm)

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Ta có:

S=131+132+133+...+160S=131+132+133+...+160

⇒S=(131+132+...+140)+(141+142+...+150)+(151+152+...+160)⇒S=(131+132+...+140)+(141+142+...+150)+(151+152+...+160)

Nhận xét:

131+132+...+140>140+140+...+140=14131+132+...+140>140+140+...+140=14

141+142+...+150>150+150+...+150=15141+142+...+150>150+150+...+150=15

151+152+...+160>160+160+...+160=16151+152+...+160>160+160+...+160=16

⇒S>14+15+16=3760>35⇒S>14+15+16=3760>35

⇒S>35(1)⇒S>35(1)

Lại có:

S=(131+132+...+140)+(141+142+...+150)+(151+152+...+160)S=(131+132+...+140)+(141+142+...+150)+(151+152+...+160)

Nhận xét:

131+132+...+140<130+130+...+130=13131+132+...+140<130+130+...+130=13

141+142+...+150<140+140+...+140=14141+142+...+150<140+140+...+140=14

151+152+...+160<150+150+...+150=15151+152+...+160<150+150+...+150=15

⇒S<13+14+15=4760<45⇒S<13+14+15=4760<45

⇒S<45(2)⇒S<45(2)

Từ (1)(1) và (2)(2)

⇒35<S<45⇒35<S<45 (Đpcm)

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để gửi lại ,

xin lỗi nhiều

hì

S sẽ có 30 số hạng. Nhóm thành 3 nhóm, mỗi nhóm 101 số hạng.

S= (1/31+1/32+...+1/40) + (1/41 + 1/42 +...+1/50) + (1/51 +1/52+...+1/60)

S < (1/30 + 1/30 +...+ 1/30) + ( 1/40 +1/40+...+1/40) + (1/50 +1/50+...+1/50)

S < 1/30 + 1/40 +1/50 ; S < 47/60 < 48/60 = 4/5 (1)

S > (1/40 + 1/40 +...=1/40) + (1/50 + 1/50 +...+1/50) + (1/60 +1/60+...+1/60)

S < 10/40 + 10/50 +10/60 ; S > 37/60 > 36/60 = 3/5 (2)

Tư (1) và (2) => 3/5 < S < 4/5