2ab(20ab^2-11)
Объяснение:
40a^2b^3-22ab привел подобные
1) Пусть n=2
![1+ \frac{1}{ \sqrt{2} } \ \textgreater \ \sqrt{2} \\ \\ \sqrt{2}* (1+ \frac{1}{ \sqrt{2} } ) \ \textgreater \ \sqrt{2} *\sqrt{2} \\ \\ \sqrt{2} +1\ \textgreater \ 2 \\ \\ \sqrt{2} \ \textgreater \ 1 \\ \\](https://tex.z-dn.net/?f=1%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D++%5C+%5Ctextgreater+%5C++%5Csqrt%7B2%7D++%5C%5C++%5C%5C++%5Csqrt%7B2%7D%2A+%281%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D+%29+%5C+%5Ctextgreater+%5C+++%5Csqrt%7B2%7D+%2A%5Csqrt%7B2%7D++%5C%5C++%5C%5C++%5Csqrt%7B2%7D+%2B1%5C+%5Ctextgreater+%5C+2+%5C%5C++%5C%5C+%5Csqrt%7B2%7D+%5C+%5Ctextgreater+%5C+1+%5C%5C++%5C%5C)
верно
2)Пусть верно при n=k
![1+ \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } +...+ \frac{1}{ \sqrt{k} } \ \textgreater \ \sqrt{k} \\ \\](https://tex.z-dn.net/?f=1%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B3%7D+%7D+%2B...%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%7D+%7D+%5C+%5Ctextgreater+%5C++%5Csqrt%7Bk%7D++%5C%5C++%5C%5C+)
3)докажем, что верно при n=k+1
![1+ \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } +...+ \frac{1}{ \sqrt{k} } + \frac{1}{ \sqrt{k+1} } \ \textgreater \ \sqrt{k}+ \frac{1}{ \sqrt{k+1} } \\ \\](https://tex.z-dn.net/?f=1%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B3%7D+%7D+%2B...%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D+%5C+%5Ctextgreater+%5C+%5Csqrt%7Bk%7D%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D++%5C%5C++%5C%5C+)
![\frac{1}{ \sqrt{k+1} } -](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D+-+)
положительное число
![\sqrt{k} + \frac{1}{ \sqrt{k+1} } \ \textgreater \ \sqrt{k+1} \\ \\ \sqrt{k+1}*( \sqrt{k} + \frac{1}{ \sqrt{k+1} } )\ \textgreater \ \sqrt{k+1} * \sqrt{k+1} \\ \\ \sqrt{k(k+1)} +1\ \textgreater \ k+1 \\ \\ \sqrt{k^2+k} \ \textgreater \ \sqrt{k^2} ;k \geq 2 \\ \\](https://tex.z-dn.net/?f=+%5Csqrt%7Bk%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D+%5C+%5Ctextgreater+%5C++%5Csqrt%7Bk%2B1%7D++%5C%5C++%5C%5C+%5Csqrt%7Bk%2B1%7D%2A%28+%5Csqrt%7Bk%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D+%29%5C+%5Ctextgreater+%5C++%5Csqrt%7Bk%2B1%7D+%2A+%5Csqrt%7Bk%2B1%7D++%5C%5C++%5C%5C++%5Csqrt%7Bk%28k%2B1%29%7D+%2B1%5C+%5Ctextgreater+%5C+k%2B1+%5C%5C++%5C%5C++%5Csqrt%7Bk%5E2%2Bk%7D+%5C+%5Ctextgreater+%5C++%5Csqrt%7Bk%5E2%7D+%3Bk+%5Cgeq+2+%5C%5C++%5C%5C+)
верно
⇒
![1+ \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{3} } +...+ \frac{1}{ \sqrt{k} } + \frac{1}{ \sqrt{k+1} } \ \textgreater \ \sqrt{k+1} } \\ \\](https://tex.z-dn.net/?f=1%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7B3%7D+%7D+%2B...%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%7D+%7D+%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bk%2B1%7D+%7D+%5C+%5Ctextgreater+%5C+%5Csqrt%7Bk%2B1%7D+%7D+%5C%5C+%5C%5C+)
ч.т.д.
(5х+3)/х так как делить на 0 нельзя
lg(х) так как под логарифмом должно стоять х>0
![\sqrt{x-3}](https://tex.z-dn.net/?f=+%5Csqrt%7Bx-3%7D+)
так как при х=0 под корнем отрицательное число
![x^{4} =(x-6) ^{2} <strong></strong> ](https://tex.z-dn.net/?f=+x%5E%7B4%7D+%3D%28x-6%29+%5E%7B2%7D+%3Cstrong%3E%3C%2Fstrong%3E++%0A%0A)
возьмем корень из левой и правой части уравнения
![\left \{ {{x^{2} =6-x} \atop {x^{2} =x-6}} \right. \\ \left \{ {{x^{2} +x-6=0} \atop {x^{2} -x+6=0}} \right. \\ \left \{ {{(x-2)(x+3)=0} \atop {D\ \textless \ 0}} \right. \\ x-2=0 \\ x=2 \\ x+3=0 \\ x=-3 \\](https://tex.z-dn.net/?f=++%5Cleft+%5C%7B+%7B%7Bx%5E%7B2%7D+%3D6-x%7D+%5Catop+%7Bx%5E%7B2%7D+%3Dx-6%7D%7D+%5Cright.+++++++%5C%5C+%0A++%5Cleft+%5C%7B+%7B%7Bx%5E%7B2%7D+%2Bx-6%3D0%7D+%5Catop+%7Bx%5E%7B2%7D+-x%2B6%3D0%7D%7D+%5Cright.+%5C%5C+%0A+%5Cleft+%5C%7B+%7B%7B%28x-2%29%28x%2B3%29%3D0%7D+%5Catop+%7BD%5C+%5Ctextless+%5C+0%7D%7D+%5Cright.+%5C%5C+%0Ax-2%3D0+%5C%5C+%0Ax%3D2+%5C%5C+%0Ax%2B3%3D0+%5C%5C+%0Ax%3D-3+%5C%5C+)
3m²+15nm-2m-10n= m +5m= 6m
a³+3a²b+ab²+3b²= a³+3a²b+3ab²+b³= (a+b)³