A=\(\left\{2;5;10;17............;10001\right\}\)
a) Viết các phần tử bằng tính chất đặc chưng
b)Tìm phần tử thứ 190
c)tính số phần tử của a
d)tính số tập con của a
e)tính tổng các phần tử của a
Những câu hỏi liên quan
ta có sqrt{left(1+a^3right)left(1+b^3right)}sqrt{left(1+aright)left(a^2-a+1right)}.sqrt{left(1+bright)left(b^2-b+1right)}
Mà sqrt{left(a+1right)left(a^2-a+1right)}ledfrac{a+1+a^2-a+2}{2}dfrac{a^2+2}{2}
Tương tự thì sqrt{left(1+a^3right)left(1+b^3right)}ledfrac{left(a^2+2right)left(b^2+2right)}{4}Rightarrowdfrac{a^2}{sqrt{left(1+a^3right)left(1+B^3right)}}gedfrac{4a^2}{left(a^2+2right)left(b^2+2right)}...
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ta có \(\sqrt{\left(1+a^3\right)\left(1+b^3\right)}=\sqrt{\left(1+a\right)\left(a^2-a+1\right)}.\sqrt{\left(1+b\right)\left(b^2-b+1\right)}\)
Mà \(\sqrt{\left(a+1\right)\left(a^2-a+1\right)}\le\dfrac{a+1+a^2-a+2}{2}=\dfrac{a^2+2}{2}\)
Tương tự thì \(\sqrt{\left(1+a^3\right)\left(1+b^3\right)}\le\dfrac{\left(a^2+2\right)\left(b^2+2\right)}{4}\Rightarrow\dfrac{a^2}{\sqrt{\left(1+a^3\right)\left(1+B^3\right)}}\ge\dfrac{4a^2}{\left(a^2+2\right)\left(b^2+2\right)}\)
=\(\dfrac{4a^2\left(c^2+2\right)}{\left(a^2+2\right)\left(b^2+2\right)\left(c^2+2\right)}\)
Tương tự rồi + vào, ta có
...\(\ge4\dfrac{a^2\left(c^2+2\right)+b^2\left(a^2+2\right)+c^2\left(b^2+2\right)}{\left(a^2+2\right)\left(b^2+2\right)\left(c^2+2\right)}\)
ta cần chứng minh \(3\left[a^2\left(c^2+2\right)+b^2\left(a^2+2\right)+c^2\left(b^2+2\right)\right]\ge\left(a^2+2\right)\left(b^2+2\right)\left(c^2+2\right)\)
đến đây nhân tung ra và dùng cô-si tiếp
Rút gọn :
a,Aleft(3+1right)left(3^2+1right)left(3^4+1right)...left(3^{64}+1right)
b,B-1^2+2^2-3^2+4^2-...-99^2+100^2
c,C-1^2+2^2-3^2+4^2-...+left(-1right)^ncdot n^2
d,D3cdotleft(2^2+1right)left(2^4+1right)...left(2^{64}+1right)+1
e,Eleft(a+b+cright)^2+left(a+b-cright)^2-2left(a+bright)^2
g,Gleft(a+b+c+dright)^2+left(a+b-c-dright)^2+left(a+c-b-dright)^2+left(a+d-b-cright)^2
h,Hleft(a+b+cright)^3-left(b+c-aright)^3-left(a+c-bright)^3+left(a+b-cright)^3
i,Ileft(a+bright)^3+left(b+cright)^3+left(c...
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Rút gọn :
\(a,A=\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\\ b,B=-1^2+2^2-3^2+4^2-...-99^2+100^2\\ c,C=-1^2+2^2-3^2+4^2-...+\left(-1\right)^n\cdot n^2\\ d,D=3\cdot\left(2^2+1\right)\left(2^4+1\right)...\left(2^{64}+1\right)+1\\ e,E=\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a+b\right)^2\\ g,G=\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\\ h,H=\left(a+b+c\right)^3-\left(b+c-a\right)^3-\left(a+c-b\right)^3+\left(a+b-c\right)^3\\ i,I=\left(a+b\right)^3+\left(b+c\right)^3+\left(c+a\right)^3-3\left(a+b\right)\left(c+b\right)\left(c+a\right)\)
Mọi người ơi, giúp mk vs, đc câu nào hay câu ấy ! Help me!!!!!!!!!!!!!!!!!!
a/ \(A=\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)
\(2A=2\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)
\(2A=\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)
\(2A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)
\(2A=\left(3^4-1\right)\left(3^4+1\right)...\left(3^{64}+1\right)\)
\(\Rightarrow2A=3^{128}-1\Rightarrow A=\dfrac{3^{128}-1}{2}\)
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e) ta dể dàng thấy được : \(a^2+b^2=\left(a+b\right)^2-2ab\)
\(\Rightarrow E=\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a+b\right)^2\)
\(=\left(2a+2b\right)^2-2\left(a+b+c\right)\left(a+b-c\right)-2\left(a+b\right)^2\)
\(=4\left(a+b\right)^2-2\left(\left(a+b\right)^2-c^2\right)-2\left(a+b\right)^2\)
\(=4\left(a+b\right)^2-2\left(a+b\right)^2+2c^2-2\left(a+b\right)^2=2c^2\)
g) củng sử dụng cái trên ta có : \(G=\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2\)
\(=\left(2a+2b\right)^2-2\left(a+b+c+d\right)\left(a+b-c-d\right)+\left(2a-2b\right)^2-2\left(a+c-b-d\right)\left(a+d-b-c\right)\)
\(=4\left(a+b\right)^2+4\left(a-b\right)^2-2\left(\left(a+b\right)^2-\left(c+d\right)^2\right)-2\left(\left(a-b\right)^2-\left(c-d\right)^2\right)\)
\(=4\left(\left(a+b\right)^2+\left(a-b\right)^2\right)-2\left(\left(a+b\right)^2+\left(a-b\right)^2\right)+2\left(\left(c+d\right)^2+\left(c-d\right)^2\right)\)
\(=2\left(\left(a+b\right)^2+\left(a-b\right)^2\right)+2\left(\left(c+d\right)^2+\left(c-d\right)^2\right)\)\(=2\left(\left(2a\right)^2-2\left(a+b\right)\left(a-b\right)\right)+2\left(\left(2c\right)^2-2\left(c+d\right)\left(c-d\right)\right)\)
\(=2\left(4a^2-2\left(a^2-b^2\right)\right)+2\left(4c^2-2\left(c^2-d^2\right)\right)\)
\(=2\left(2a^2+2b^2\right)+2\left(2c^2+2d^2\right)=4\left(a^2+b^2+c^2+d^2\right)\)
bn đăng nhiều quá nên mk làm câu nào hay câu đó nha
mà nè mấy câu a;b;c;d hình như trên mạng có bn lên đó tìm nha .
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Chặt hơn một bài toán quen thuộc :3Với a, b, c là các số thực:a^2+b^2+c^2-ab-bc-cagefrac{Sigma a^2left(a-bright)left(a-cright)}{left(a+b+cright)^2}ge0Hôm ngồi vọc Maple:left(Sigma a^2-Sigma abright)left[Sigma a^2left(a-bright)left(a-cright)right]left[Sigma aleft(a-bright)left(a-cright)right]^2+3left(a-bright)^2left(b-cright)^2left(c-aright)^2Có ai so sánh giúp mình 2 bất đẳng thức: left{left[Sigma aleft(a-bright)left(a-cright)right]^2+3left(a-bright)^2left(b-cright)^2left(c-aright)^2right}left(a...
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Chặt hơn một bài toán quen thuộc :3
Với a, b, c là các số thực:
\(a^2+b^2+c^2-ab-bc-ca\ge\frac{\Sigma a^2\left(a-b\right)\left(a-c\right)}{\left(a+b+c\right)^2}\ge0\)
Hôm ngồi vọc Maple:
\(\left(\Sigma a^2-\Sigma ab\right)\left[\Sigma a^2\left(a-b\right)\left(a-c\right)\right]=\left[\Sigma a\left(a-b\right)\left(a-c\right)\right]^2+3\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2\)
Có ai so sánh giúp mình 2 bất đẳng thức: \(\left\{\left[\Sigma a\left(a-b\right)\left(a-c\right)\right]^2+3\left(a-b\right)^2\left(b-c\right)^2\left(c-a\right)^2\right\}\left(a+b+c\right)^2\) và \(\left(\Sigma a^2\left(a-b\right)\left(a-c\right)\right)^2\) vế nào lớn hơn được không?
M=\(\dfrac{a\left(1+b^2\right)\left(1+c^2\right)}{\left(1+a^2\right)\left(b+c\right)}+\dfrac{b\left(1+c^2\right)\left(1+a^2\right)}{\left(1+b^2\right)\left(a+c\right)}+\dfrac{c\left(1+a^2\right)\left(1+b^2\right)}{\left(1+c^2\right)\left(a+b\right)}\)
65. Phân tích đa thức thành nhân tửa) ableft(a+bright)-bcleft(b+cright)+acleft(a-cright)b) aleft(b^2+c^2right)+bleft(c^2+a^2right)+cleft(a^2+b^2right)+2abcc) left(a+bright)left(a^2-b^2right)+left(b+cright)left(b^2+c^2right)+left(c+aright)left(c^2+a^2right)d) a^3left(b-cright)+b^3left(c-aright)+c^3left(a-bright)e) a^3left(c-b^2right)+b^3left(a-c^2right)+c^3left(b-a^2right)+abcleft(abc-1right)
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65. Phân tích đa thức thành nhân tử
a) \(ab\left(a+b\right)-bc\left(b+c\right)+ac\left(a-c\right)\)
b) \(a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)+2abc\)
c) \(\left(a+b\right)\left(a^2-b^2\right)+\left(b+c\right)\left(b^2+c^2\right)+\left(c+a\right)\left(c^2+a^2\right)\)
d) \(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
e) \(a^3\left(c-b^2\right)+b^3\left(a-c^2\right)+c^3\left(b-a^2\right)+abc\left(abc-1\right)\)
65. Phân tích đa thức thành nhân tửa) ableft(a+bright)-bcleft(b+cright)+acleft(a-cright)b) aleft(b^2+c^2right)+bleft(c^2+a^2right)+cleft(a^2+b^2right)+2abcc) left(a+bright)left(a^2-b^2right)+left(b+cright)left(b^2-c^2right)+left(c+aright)left(c^2-a^2right)d) a^3left(b-cright)+b^3left(c-aright)+c^3left(a-bright)e) a^3left(c-b^2right)+b^3left(a-c^2right)+c^3left(b-a^2right)+abcleft(abc-1right)
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65. Phân tích đa thức thành nhân tử
a) \(ab\left(a+b\right)-bc\left(b+c\right)+ac\left(a-c\right)\)
b) \(a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)+2abc\)
c) \(\left(a+b\right)\left(a^2-b^2\right)+\left(b+c\right)\left(b^2-c^2\right)+\left(c+a\right)\left(c^2-a^2\right)\)
d) \(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
e) \(a^3\left(c-b^2\right)+b^3\left(a-c^2\right)+c^3\left(b-a^2\right)+abc\left(abc-1\right)\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
\(B=\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
\(=\left(\dfrac{a-b}{a\left(a+b\right)}-\dfrac{a}{b\left(a+b\right)}\right):\left(\dfrac{b^3}{a\left(a-b\right)\left(a+b\right)}+\dfrac{1}{a+b}\right)\)
\(=\dfrac{b\left(a-b\right)-a^2}{ab\left(a+b\right)}:\dfrac{b^3+a\left(a-b\right)}{a\left(a-b\right)\left(a+b\right)}\)
\(=\dfrac{ab-b^2-a^2}{ab\left(a+b\right)}\cdot\dfrac{a\left(a-b\right)\left(a+b\right)}{a^2-ab+b^3}\)
\(=\dfrac{\left(a-b\right)\left(ab-b^2-a^2\right)}{b\left(a^2-ab+b^3\right)}\)
\(=\dfrac{-\left(a-b\right)\left(a^2-ab+b^2\right)}{b\left(a^2-ab+b^3\right)}\)
Đề lỗi rồi chứ mình ko rút gọn đc nữa
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Qleft(frac{2}{2+2sqrt{a}}+frac{1}{2-2sqrt{a}}-frac{a^2+1}{1-a^2}right)left(1+frac{1}{a}right)left(frac{1}{2left(1+sqrt{a}right)}+frac{1}{2left(1-sqrt{a}right)}-frac{a^2+1}{left(1-sqrt{a}right)left(1+sqrt{a}right)left(1+aright)}right)left(frac{a+1}{a}right)left(frac{left(1-sqrt{a}right)left(1+aright)+left(1+sqrt{a}right)left(1+aright)-2left(a^2+1right)}{2left(1-aright)left(1+aright)}right)left(frac{a+1}{a}right)left(frac{1+a-sqrt{a}-asqrt{a}+1+a+sqrt{a}+asqrt{a}-2a^2-2}{2left(1-aright)left(1+arig...
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\(Q=\left(\frac{2}{2+2\sqrt{a}}+\frac{1}{2-2\sqrt{a}}-\frac{a^2+1}{1-a^2}\right)\left(1+\frac{1}{a}\right)\)
\(=\left(\frac{1}{2\left(1+\sqrt{a}\right)}+\frac{1}{2\left(1-\sqrt{a}\right)}-\frac{a^2+1}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(1+a\right)}\right)\left(\frac{a+1}{a}\right)\)
\(=\left(\frac{\left(1-\sqrt{a}\right)\left(1+a\right)+\left(1+\sqrt{a}\right)\left(1+a\right)-2\left(a^2+1\right)}{2\left(1-a\right)\left(1+a\right)}\right)\left(\frac{a+1}{a}\right)\)
\(=\left(\frac{1+a-\sqrt{a}-a\sqrt{a}+1+a+\sqrt{a}+a\sqrt{a}-2a^2-2}{2\left(1-a\right)\left(1+a\right)}\right)\left(\frac{a+1}{a}\right)\)
\(=\left(\frac{2a-2a^2}{2\left(1-a\right)\left(1+a\right)}\right)\)
\(=\frac{a}{a}\)= 1