chứng minh rằng : S=1/1.2+1/2.3+...+1/2014.2015<1
Tinh tong S =1/1.2+1/2.3+1/3.4+...+1/2014.2015+1/2015.2016
\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.......+\frac{1}{2015.2016}\)
\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.......+\frac{1}{2015}-\frac{1}{2016}\)
\(S=\frac{1}{1}-\frac{1}{2016}=\frac{2015}{2016}\)
S=1/1-1//2+1/2-1/3+1/3-1/4+.......=1/2014-1/2015
S=1/1-1/2015
S=2015/2015-1/2015
S=2014/2015
Chứng minh rằng :s=1/1.2+1/2.3+...+1/n.(n+1)
s ko phải số tự nhiên
ta có: \(\frac{1}{1.2}>0;\frac{1}{2.3}>0;...;\frac{1}{n.\left(n+1\right)}>0\)
\(\Rightarrow S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{n.\left(n+1\right)}>0\)
ta có: \(S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{n.\left(n+1\right)}\)
\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n}-\frac{1}{n+1}\)
\(S=1-\frac{1}{n+1}< 1\)
=> 0 < S < 1
=> S không phải là số tự nhiên
Tính nhanh
C=1.2+2.3+3.4+...+2014.2015
K=1.2+2.3+3.4+..+(n-1).n
cau hỏi tương tự ko có mà!!!!!!!!!!!!!!!!!!!!!!!!!!!!
3C=1.2.3+2.3.(4-1)+3.4.(5-2)+...+2014.2015.(2016-2013)
3C=2014.2015.2016
C=2014.2015.2016:3
D=1(1+2+3+......+2014)+(1.2+2.3+3.4+.....+2014.2015)
Đặt A = 1 + 2 + 3 + 4 + ....... + 2014
Số các số hạng của A là:
2014 - 1 + 1 = 2014 (số)
A = 2014.(2014 + 1):2 = 2029105
Đặt B = 1.2 + 2.3 + 3. 4 + .....+ 2014.2015
3B = 1.2.3 + 2.3.3 + .3.4.3 + ......... + 2014.2015
3B = 1.2.3 + 2.3.(4-1) + 3.4.(5-2) + .......... + 2014.2015.(2016 - 2013)
3B = 1.2.3 + 2.3.4 - 1.2.3 + ........+2014.2015.2016 - 2013.2014.2015
3B = 2014 . 2015 .2016 = 8181351360
B = 8181351360 : 3
B = 2727117120
Vậy D = A + B = 2029105 + 2727117120 = 2729146225
Chứng minh rằng :S=1/1.2+1/2.3+...+1/n.(n+1)
ko phải số tự nhiên
Đk: n khác 0, n khác -1
\(S=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{n\left(n+1\right)}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n}-\dfrac{1}{n+1}\)
\(=1-\dfrac{1}{n+1}\)
Vì \(0< \dfrac{1}{n+1}< 1\) (n khác 0, n khác -1) nên \(0< 1-\dfrac{1}{n+1}< 1\)
hay 0<S<1
Vậy S không là stn
cho bieu thuc 1/1.2+1/2.3...+1/2014.2015. Gia tri thoa man A+1/2015=2x
A= 1-1/2015=2014/2015
=> 2014/2015+1/2015=2x
=> 1=2x
=> x=1/2
chứng minh rằng 1/1.2 + 1/2.3 + 1/3.4+ 1/4.5+ ...+1/49.50 <1
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{49.50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{1}{50}< 1\) (đpcm)
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{49.50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{1}{50}\)
\(=\frac{49}{50}\)
\(\Rightarrow\) Quy đồng phân số và 1 là : \(\frac{49}{50}\) và \(1\)
Giữ nguyên phân số \(\frac{49}{50}\)
Ta có : \(\frac{1}{1}=\frac{1.50}{1.50}=\frac{50}{50}\)
\(\Rightarrow\frac{49}{50}< \frac{50}{50}\left(đpcm\right)\)
Cho S=\(\frac{1}{1.2}\)+\(\frac{1}{1.2+2.3}\)+...+\(\frac{1}{1.2+2.3+3.4+...+n.\left(n+1\right)}\)
Chứng minh S<\(\frac{3}{4}\)
Ta có: \(\frac{1}{1.2}=\frac{3}{1.2.3}\) ;\(\frac{1}{1.2+2.3}=\frac{3}{2.3.4}\); \(\frac{1}{2.3+3.4}=\frac{3}{3.4.5}\); ......;\(\frac{1}{1.2+2.3+3.4+...+n\left(n+1\right)}=\frac{3}{n\left(n+1\right)\left(n+2\right)}\)
=> \(S=\frac{3}{1.2.3}+\frac{3}{2.3.4}+\frac{3}{3.4.5}+...+\frac{3}{n\left(n+1\right)\left(n+2\right)}\)
=> \(\frac{2S}{3}=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{n\left(n+1\right)\left(n+2\right)}\)
Ta lại có: \(\frac{2}{1.2.3}=\frac{1}{1.2}-\frac{1}{2.3}\); \(\frac{2}{2.3.4}=\frac{1}{2.3}-\frac{1}{3.4}\); \(\frac{2}{3.4.5}=\frac{1}{3.4}-\frac{1}{4.5}\);....;\(\frac{2}{n\left(n+1\right)\left(n+2\right)}=\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\)
=> \(\frac{2S}{3}=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\)
=> \(\frac{2S}{3}=\frac{1}{1.2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\)=> \(S=\frac{3}{4}-\frac{3}{2\left(n+1\right)\left(n+2\right)}< \frac{3}{4}\)
=> \(S< \frac{3}{4}\)
Mình nhầm 1 chỗ: \(\frac{1}{1.2+2.3+3.4}=\frac{3}{3.4.5}\)
Tinh A = \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)
\(=1-\frac{1}{2015}\)
\(=\frac{2014}{2015}\)
A=1/1-1/2+1/2-1/3+1/3-...........+1/2014-1/2015
A=1/1-1/2015
A=2014/2015