Đại số tổng hợp

Cmr: $1) 2.1C^{2}_{n}+3.2C^{3}_{n}+4.3C^{4}_{n}+...+n(n-1)C^{n}_{n}=n(n-1)2^{n-2}$2) $(C^{0}_{2n})^{2}-(C^{1}_{2n})^{2}(x^{2}_{2n})^{2}-...+(C^{2n}_{2n})^{2}=(-1)^{n}C^{n}_{2n}$

Tổ hợp

Đại số tổng hợp

Cmr: $1) 2.1C^{2}_{n}+3.2C^{3}_{n}+4.3C^{4}_{n}+...+n(n-1)C^{n}_{n}=n(n-1)2^{n-2}$2) $(C^{0}_{2n})^{2}-(C^{1}_{2n})^{2}(C^{2}_{2n})^{2}-...+(C^{2n}_{2n})^{2}=(-1)^{n}C^{n}_{2n}$

Tổ hợp

cosx + cos3x + 2cos5x=0<=>$4cos^{3}x -2cosx + 2cos(3x+2x)=0$<=>$2cosx(2cos^{2}-1)+2cos3xcos2x- 2sin3xsin2x=0$<=>$cosxcos2x+cos3xcos2x-sin3xsin2x=0$<=>$cos2x(cosx +cos3x)-sin3xsin2x=0$<=>$2cos^{2}2xcosx- 2sinxcosxsin3x=0$<=> $2cosx(cos^{2}2x- sinxsin3x)= 0$<=>$\begin{cases}cosx=0 <=> x=\frac{\pi }{2}+k \pi \\ cos^{2}2x -sinxsin3x=0(*)\end{cases}$$(*)<=> cos^{2}2x-\frac{1}{2}(cos2x-cos4x)=0$<=>$cos^{2}2x -\frac{1}{2}cos2x+cos^{2}2x-\frac{1}{2}=0$<=>$4cos^{2}2x - coss2x -1=0$

cosx + cos3x + 2cos5x=0<=>$4cos^{3}x -2cosx + 2cos(3x+2x)=0$<=>$2cosx(2cos^{2}-1)+2cos3xcos2x- 2sin3xsin2x=0$<=>$cosxcos2x+cos3xcos2x-sin3xsin2x=0$<=>$cos2x(cosx +cos3x)-sin3xsin2x=0$<=>$2cos^{2}2xcosx- 2sinxcosxsin3x=0$<=> $2cosx(cos^{2}2x- sinxsin3x)= 0$<=>$\begin{cases}cosx=0 <=> x=\frac{\pi }{2}+k \pi \\ cos^{2}2x -sinxsin3x=0(*)\end{cases}$$(*)<=> cos^{2}2x-\frac{1}{2}(cos2x-cos4x)=0$<=>$cos^{2}2x -\frac{1}{2}cos2x+cos^{2}2x-\frac{1}{2}=0$<=>$4cos^{2}2x - cos2x -1=0$

$B=10+10^{2}+...+10^{n}+5+55+555+...+555...55$$=10+10^{2}+...+10^{n+}\frac{5}{9}(9+99+999+...+999...99)$$=10+10^{2}+...+10^{n+}\frac{5}{9}(10+10^{2}+...+10^{n}-n)$$=\frac{14}{9}(10+10^{2}+...+10^{n})-\frac{5}{9}n=\frac{14}{9}(\frac{10(10^{n}-1)}{9}-n)-\frac{5}{9}n$

$B=10+10^{2}+...+10^{n}+5+55+555+...+555...55$$=10+10^{2}+...+10^{n}+\frac{5}{9}(9+99+999+...+999...99)$$=10+10^{2}+...+10^{n}+\frac{5}{9}(10+10^{2}+...+10^{n}-n)$$=\frac{14}{9}(10+10^{2}+...+10^{n})-\frac{5}{9}n=\frac{14}{9}(\frac{10(10^{n}-1)}{9}-n)-\frac{5}{9}n$