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TfBoyS_TDT
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soyeon_Tiểubàng giải
11 tháng 9 2016 lúc 12:55

\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(=1-\frac{1}{2!}+1-\frac{1}{3!}+\frac{1}{2!}-\frac{1}{4!}+...+\frac{1}{98!}-\frac{1}{100!}\)

\(=\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(=2-\frac{1}{99!}-\frac{1}{100!}< 2\left(đpcm\right)\)

Kaori Miyazono
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Phạm Tuấn Đạt
3 tháng 9 2017 lúc 22:37

Ta xét :

\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(=\left(1+1+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\)

\(=2-\frac{1}{99}-\frac{1}{100}\)

Mà \(2-\frac{1}{99}-\frac{1}{100}< 2\)

\(\RightarrowĐPCM\)

le ngoc hieu
28 tháng 8 2017 lúc 20:26

 1.2−12! +2.3−13! +3.4−14! +....+99.100−1100=2 suy ra 1.2−12! +2.3−13! +3.4−14! +....+99.100−1100<2

zoombie hahaha
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Hồ Thu Giang
5 tháng 9 2015 lúc 16:24

 

\(\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(\left(2+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(2-\frac{1}{99!}-\frac{1}{100!}

Game Master VN
30 tháng 6 2018 lúc 7:30

tớ là một youtuber link đây https://www.youtube.com/channel/UCRoT6fvb0VTS8S1EFsH0qGg?sub_confimation=1 nhớ đăng ký, , chia sẻ ủng hộ giúp mình nhé

kiwi nguyễn
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Nguyễn Kim Hưng
26 tháng 6 2019 lúc 10:30

a)\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}\)

=\(\frac{2}{2!}-\frac{1}{2!}+\frac{3}{3!}-\frac{1}{3!}+\frac{4}{4!}-\frac{1}{4!}+...+\frac{100}{100!}-\frac{1}{100!}\)

=\(1-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...+\frac{1}{99!}-\frac{1}{100!}\)

=\(1-\frac{1}{100!}< 1\)

\(\Rightarrow\)\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}< 1\)

b)\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

=\(\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

=\(\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)=\(1+1-\frac{1}{99}-\frac{1}{100}\)

=\(2-\frac{1}{99}-\frac{1}{100}< 2\)

\(\Rightarrow\)\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}< 2\)

zoombie hahaha
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nguyen thi huyen
5 tháng 9 2015 lúc 15:42

ta có:

1.2-1/2!+2.3-1/3!+3.4-1/4!+...+99.100-1/100!

=1.2/2!-1/2!+2.3/3!-13!+...+99.100-1/100!

=(1.2/2!+2.3/3!+3.4-4!+...+99.100/100!)-(1/2!+1/3!+...+1/100!)

=(1+1+1/2+...+1/98!)_(1/2!+1/3!+...+1/100!)

=2-1/99!-1/100!<2

Phạm Tuấn Đạt
12 tháng 9 2017 lúc 23:24

Ta xét :

\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(=1+1-\frac{1}{99}-\frac{1}{100}\)

\(=2-\frac{1}{99}-\frac{1}{100}< 2\)

\(\RightarrowĐPCM\)

Doraemon
15 tháng 9 2018 lúc 9:03

Ta có:

\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(=1+1-\frac{1}{99}-\frac{1}{100}\)

\(=2-\frac{1}{99}-\frac{1}{100}< 2\)

Vậy \(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}< 2\)

Duong Thi Nhuong
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Sorano Yuuki
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Tạ Giang Thùy Loan
2 tháng 6 2017 lúc 18:05

\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}\)\(+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)\)\(-\frac{1}{2!}-\frac{1}{3!}-\frac{1}{4!}-...-\frac{1}{100!}\)

\(=1+1+\frac{1}{2!}+...+\frac{1}{98!}-\frac{1}{2!}-\frac{1}{3!}-\frac{1}{4!}-...-\frac{1}{100!}\)

\(=2-\frac{1}{99!}-\frac{1}{100!}< 2\)

Trà My
2 tháng 6 2017 lúc 16:45

\(=1-\frac{1}{2!}+\frac{1}{1!}-\frac{1}{3!}+\frac{1}{2!}-\frac{1}{4!}+...+\frac{1}{98!}-\frac{1}{100!}\)

\(=2-\frac{1}{99!}-\frac{1}{100!}\)

Đặng Quốc Huy
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Vũ Minh Tuấn
6 tháng 1 2020 lúc 18:59

Đặt \(A=\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)

\(\Rightarrow A=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)

\(\Rightarrow A=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(\Rightarrow A=\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{\text{4!}}+...+\frac{1}{100!}\right)\)

\(\Rightarrow A=1+1-\frac{1}{99!}-\frac{1}{100!}\)

\(\Rightarrow A=2-\frac{1}{99!}-\frac{1}{100!}\)

\(2-\frac{1}{99!}-\frac{1}{100!}< 2.\)

\(\Rightarrow A< 2\left(đpcm\right).\)

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