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\(y=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)....\left(\frac{1}{2014^2}-1\right)\)
\(y=\left(\frac{-1.3}{2.2}\right)\left(\frac{-2.4}{3.3}\right)....\left(\frac{-2013.2015}{2014.2014}\right)\)
\(y=-\left(\frac{1.2....2013.3.4...2015}{2.3....2014.2.3....2014}\right)\)
\(y=-\left(\frac{2015}{2014.2}\right)\)
\(y=\frac{-2015}{4028}\)
\(x=\frac{-1}{2}=\frac{-2014}{4028}\)
Vì \(\frac{-2015}{4028}<\frac{-2014}{4028}\)
=> y < x
Tính ra A là 2-(1/2)^2013. Phần còn lại thì quá dễ r
(Để tính A từ dãy trên ta nhân 2 lên thành 2A. Rồi lấy 2A-A=A=...)
\(A=1+\frac{1}{2}+\left(\frac{1}{2}\right)^2+..............+\left(\frac{1}{2}\right)^{2013}\)
\(\Rightarrow2A=2+1+\frac{1}{2}+.......+\left(\frac{1}{2}\right)^{2013}\Rightarrow2A-A=A=2-\left(\frac{1}{2}\right)^{2013}\)
\(VI:A+\left(\frac{1}{2}\right)^n=2\Rightarrow n=2013\)
\(A=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{2014}\)
\(A=\frac{1}{2}+\frac{1^2}{2^2}+...+\frac{1^{2014}}{2^{2014}}\)
\(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\)
\(2A=2\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\right)\)
\(2A=1+\frac{1}{2}+...+\frac{1}{2^{2013}}\)
\(2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^{2013}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2014}}\right)\)
\(A=1-\frac{1}{2^{2014}}< 1\)
Đpcm
\(A=\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right)....\left(\frac{1}{100^2}-1\right)\)( có 2013 thừa số )
\(A=\left(-\frac{3}{2^2}\right).\left(-\frac{8}{3^2}\right).\left(-\frac{15}{4^2}\right).....\left(-\frac{\text{4056196}}{2014^2}\right)\)
\(-A=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.....\frac{4056196}{2014^2}=\frac{1.3.2.4.3.5....2013.2015}{2.2.3.3.4.4.....2014.2014}\)
\(-A=\frac{\left(1.2.3...2013\right).\left(3.4.5.6...2015\right)}{\left(2.3.4.5....2014\right).\left(2.3.4.5...2014\right)}=\frac{1.2015}{2.2014}=\frac{2015}{4028}\)
\(A=-\frac{2015}{4028}\)
Vậy.....
-A=(\(1-\frac{1}{2^2}\)) . (\(1-\frac{1}{3^2}\))......(\(1-\frac{1}{2014^2}\))
-A= \(\frac{3}{4}\). \(\frac{8}{9}\). ...... \(\frac{4056195}{4056196}\)
-A= \(\frac{1.3.2.4.......2013.2015}{2.2.3.3.......2.14.2014}\)
-A= \(\frac{\left(1.2.3...2013\right)\left(3.4.5...2015\right)}{\left(2.3.4...2014\right)\left(2.3.4...2014\right)}\)
-A= \(\frac{2015}{2014.2}\)
-A=\(\frac{2015}{4028}\)
Bài 1
\(a,\left(\frac{3}{5}\right)^2-\left[\frac{1}{3}:3-\sqrt{16}.\left(\frac{1}{2}\right)^2\right]-\left(10.12-2014\right)^0\)
\(=\frac{9}{25}-\left[\frac{1}{9}-4.\frac{1}{4}\right]-1\)
\(=\frac{9}{25}-\left(-\frac{8}{9}\right)-1\)
\(=\frac{9}{25}+\frac{8}{9}-1\)
\(=\frac{56}{225}\)
\(b,|-\frac{100}{123}|:\left(\frac{3}{4}+\frac{7}{12}\right)+\frac{23}{123}:\left(\frac{9}{5}-\frac{7}{15}\right)\)
\(=\frac{100}{123}:\left(\frac{4}{3}\right)+\frac{23}{123}:\frac{4}{3}\)
\(=\left(\frac{100}{123}+\frac{23}{123}\right):\frac{4}{3}\)
\(=1:\frac{4}{3}=\frac{3}{4}\)
Phần c đăng riêng vì mk chưa tìm đc cách giải bt mỗi đáp án :v
\(c,\frac{\left(-5\right)^{32}.20^{43}}{\left(-8\right)^{29}.125^{25}}\)
\(=\frac{\left(-5\right)^{32}.\left(4.5\right)^{43}}{\left[4.\left(-2\right)\right]^{29}.\left(-5^3\right)^{25}}\)
\(=\frac{-5^{32}.4^{43}.5^{43}}{4^{29}.\left(-2\right)^{29}.\left(5\right)^{75}}\)
\(=\frac{\left(-5^4\right)^8.4^{43}.5^{43}}{4^{29}.\left(-2\right)^{29}.\left(5^3\right)^{25}}\)
\(=-\frac{1}{2}\)
Vì \(\frac{1}{2^2}>0\)
............
\(\frac{1}{2014^2}>0\)
=> A = \(\left(\frac{1}{2^2}\right)\left(\frac{1}{3^2}\right)...\left(\frac{1}{2014^2}\right)>0\)
B = \(-\frac{1}{2}<0\)
Vậy A > B
gap A len 1/2
\(2A=1+\frac{1}{2}+\left(\frac{1}{2}\right)^2+...+\left(\frac{1}{2}\right)^{2015}\)
\(\Rightarrow2A-A=1-\left(\frac{1}{2}\right)^{2014}\Rightarrow A=1-\left(\frac{1}{2}\right)^{2014}< 1\)