CMR : B = 1/2^2 + 1/3^2 + 1/4^2 + .....+ 1/2016^2 < 3/4
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
a)\(M=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2016^2}<1\)
\(\Rightarrow2M=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}<1\)
\(2M-M=\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}\right)-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2016^2}\right)<1\)
\(\Rightarrow M=1-\frac{1}{2016^2}\)<1
=>(DPCM)
CÂU b và c làm tương tự
\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2016^2}\)
\(=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{2016.2016}\)
\(< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2015.2016}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2015}-\frac{1}{2016}\)
\(=1-\frac{1}{2016}\)
\(=\frac{2015}{2016}< 1\)
\(\Rightarrow A< 1\)
\(\text{Vậy }A< 1\left(\text{đpcm}\right)\)
Bài giải
Ta có : \(\frac{1}{2^2}< \frac{1}{1\cdot2}\) ; \(\frac{1}{3^2}< \frac{1}{2\cdot3}\) ; \(\frac{1}{4^2}< \frac{1}{3\cdot4}\) ; ... ; \(\frac{1}{2016^2}< \frac{1}{2015\cdot2016}\)
\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2016^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2015\cdot2016}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2015}-\frac{1}{2016}=1-\frac{1}{2016}=\frac{2015}{2016}< 1\)
\(\Rightarrow\text{ }A< 1\)
Ta có :
\(B=\frac{2016}{1}+\frac{2015}{2}+\frac{2014}{3}+...+\frac{1}{2016}\)
\(B=\left(\frac{2015}{2}+1\right)+\left(\frac{2014}{3}+1\right)+...+\left(\frac{1}{2016}+1\right)+1\)
\(B=\frac{2017}{2}+\frac{2017}{3}+...+\frac{2017}{2016}+\frac{2017}{2017}\)
\(B=2017.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2016}+\frac{1}{2017}\right)\)
\(\Rightarrow\frac{B}{A}=\frac{2017.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2016}+\frac{1}{2017}\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2017}}=2017\)
Vậy \(\frac{B}{A}\)là số nguyên
(a - b)2 \(\ge0\Leftrightarrow a^2+b^2-2ab\ge0\Leftrightarrow a^2+b^2\ge2ab\)
=> \(\frac{1}{a^2+b^2}< \frac{1}{2ab}\left(a;b>0;a\ne b\right)\)
Áp dụng vào bài toán ta có:
\(\frac{1}{1^2+2^2}+\frac{1}{2^2+3^2}+\frac{1}{3^2+4^2}+...+\frac{1}{2016^2+2017^2}< \frac{1}{2}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2016.2017}\right)\)
\(< \frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2016}-\frac{1}{2017}\right)\)
\(< \frac{1}{2}\left(1-\frac{1}{2017}\right)< \frac{1}{2}\left(đpcm\right)\)
nhóm cuối sẽ nhóm được thành nhiều nhóm:
(1/2016+2015/2016)+(2/2016+2014/2016)+.......+(1008/2016+1008/2016) có tổng cộng 1008 nhóm =1
suy ra nhóm trên có kq là 1008
= 1/2+1+1+1008
=1/2+1010
=2021/2
cho mik nha
(1/2016+2015/2016)+(2/2016+2014/2016)+.......+(1008/2016+1008/2016) có tổng cộng 1008 nhóm =1
suy ra nhóm trên có kq là 1008
= 1/2+1+1+1008
=1/2+1010
=2021/2