Chứng minh
M=\(\frac{1}{1+2+3}\)+\(\frac{1}{1+2+3+4}\)+\(\frac{1}{1+2+3+4+5}\)+.....+\(\frac{1}{1+2+3+....+59}\)<\(\frac{2}{3}\)
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(\frac{1}{M}=\frac{1}{\frac{3.4}{2}}+\frac{1}{\frac{4.5}{2}}+...+\frac{1}{\frac{59.60}{2}}\)
\(\frac{1}{M}=\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{59.60}\)
\(\frac{1}{M}=2.\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....+\frac{1}{59}-\frac{1}{60}\right)\)
\(\frac{1}{M}=\frac{2}{3}-\frac{2}{60}< \frac{2}{3}\)
-theo t đề là M chứ ko phải 1/M
\(\frac{1}{M}=\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+...+\frac{1}{1+2+3+...+59}\)
\(\frac{1}{M}=\frac{1}{3\left(1+3\right):2}+\frac{1}{4\left(1+4\right):2}+\frac{1}{5\left(1+5\right):2}+...+\frac{1}{59\left(1+59\right):2}\)
\(\frac{1}{M}=\frac{2}{3.4}+\frac{2}{4.5}+\frac{2}{5.6}+...+\frac{2}{59.60}\)
\(\frac{1}{M}=2\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{59}-\frac{1}{60}\right)\)
\(\frac{1}{M}=2\left(\frac{1}{3}-\frac{1}{60}\right)\)
\(\frac{1}{M}=\frac{1}{2}.\frac{19}{60}\)
\(\frac{1}{M}=\frac{19}{120}\)
\(M=\frac{120}{19}>\frac{2}{3}\left(đpcm\right)\)
Mik lười quá bạn tham khảo câu 3 tại đây nhé:
Câu hỏi của nguyen linh nhi - Toán lớp 6 - Học toán với OnlineMath
\(S=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{37\cdot38\cdot39}\)
\(2S=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+...+\frac{1}{37\cdot38}-\frac{1}{38\cdot39}\)
\(2S=\frac{1}{2}-\frac{1}{38\cdot39}\)
\(S=\frac{1}{4}-\frac{1}{2\cdot38\cdot39}< \frac{1}{4}\)
Ta có : \(\frac{1}{m}=\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{59.60}=2\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{59}-\frac{1}{60}\right)\)
\(=2\left(\frac{1}{3}-\frac{1}{60}\right)=\frac{19}{30}\)
\(\Rightarrow m=\frac{30}{19}>\frac{2}{3}\)
\(Tac\text{ó}:\frac{1}{m}=\frac{2}{3.4}+\frac{2}{4.5}+.....+\frac{2}{59.60}=2\left(\frac{1}{3}-\frac{1}{4}+.......+\frac{1}{59}-\frac{1}{60}\right)\)
\(=>2\left(\frac{1}{3}-\frac{1}{60}\right)=\frac{19}{30}\\ =>m=\frac{30}{19}>\frac{2}{3}\)
\(\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+59}\)
\(=\frac{1}{\frac{3.4}{2}}+\frac{1}{\frac{4.5}{2}}+...+\frac{1}{\frac{59.60}{2}}\)
\(=\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{59.60}\)
\(=2.\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{59.60}\right)\)
\(=2.\left(\frac{4-3}{3.4}+\frac{5-4}{4.5}+...+\frac{60-59}{59.60}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{59}-\frac{1}{60}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{60}\right)\)
\(=2.\frac{19}{60}\)
\(=\frac{38}{60}\)\(< \frac{40}{60}=\frac{2}{3}\)