RiemannSums

3105 days ago by MATH5HBC2012

#1) show: 1+2+3+...+n = n(n+1)/2 #1) let: 1+2+3+...+n = a*n^2+b*n+c #1) if n=0: 0 == c #1) if n=1: 1 == a+b+c #1) if n=2: 3 == 4*a+2*b+c var('a,b,c') show(solve([0 == c,1 == a+b+c,3 == 4*a+2*b+c],(a,b,c))) var('n') show(factor(n^2/2+n/2)) var('y') show(solve(1==x+y,y)) show(solve(3==4*x+2*y,y)) plot([-x+1,-2*x+3/2]) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{2}\right), b = \left(\frac{1}{2}\right), c = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, {\left(n + 1\right)} n
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -x + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -2 \, x + \frac{3}{2}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{2}\right), b = \left(\frac{1}{2}\right), c = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, {\left(n + 1\right)} n
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -x + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -2 \, x + \frac{3}{2}\right]
#2) show: 1^2+2^2+3^2+...+n^2 = n(n+1)(2n+1)/6 #2) let: 1^2+2^2+3^2+...+n^2 = a*n^3+b*n^2+c*n+d #2) if n=0: 0==d #2) if n=1: 1==a+b+c+d #2) if n=2: 5==8*a+4*b+2*c+d #2) if n=3: 14==27*a+9*b+3*c+d var('d') show(solve([0==d,1==a+b+c+d,5==8*a+4*b+2*c+d,14==27*a+9*b+3*c+d],(a,b,c,d))) show(factor(n^3/3+n^2/2+n/6)) var('z') show(solve(1==x+y+z,z)) show(solve(5==8*x+4*y+2*z,z)) show(solve(14==27*x+9*y+3*z,z)) p1=plot3d(-x-y+1,(x,0,1),(y,0,1),color='red') p2=plot3d(-4*x-2*y+5/2,(x,0,1),(y,0,1)) p3=plot3d(-9*x-3*y+14/3,(x,0,1),(y,0,1),color='green') (p1+p2+p3).show(viewer='tachyon') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{3}\right), b = \left(\frac{1}{2}\right), c = \left(\frac{1}{6}\right), d = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6} \, {\left(n + 1\right)} {\left(2 \, n + 1\right)} n
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -x - y + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -4 \, x - 2 \, y + \frac{5}{2}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -9 \, x - 3 \, y + \frac{14}{3}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{3}\right), b = \left(\frac{1}{2}\right), c = \left(\frac{1}{6}\right), d = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6} \, {\left(n + 1\right)} {\left(2 \, n + 1\right)} n
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -x - y + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -4 \, x - 2 \, y + \frac{5}{2}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -9 \, x - 3 \, y + \frac{14}{3}\right]
#2) show: 1^3+2^3+3^3+...+n^3 = (n(n+1)/2)^2 #2) let: 1^3+2^3+3^3+...+n^3 = a*n^4+b*n^3+c*n^2+d*n+e #2) if n=0: 0==e #2) if n=1: 1==a+b+c+d+e #2) if n=2: 9==16*a+8*b+4*c+2*d+e #2) if n=3: 36==81*a+27*b+9*c+3*d+e #2) if n=4: 100==256*a+64*b+16*c+4*d+e var('e') show(solve([0==e,1==a+b+c+d+e,9==16*a+8*b+4*c+2*d+e,36==81*a+27*b+9*c+3*d+e,100==256*a+64*b+16*c+4*d+e],(a,b,c,d,e))) show(factor(n^4/4+n^3/2+n^2/4)) show(solve(1==x+y+z,z)) show(solve(9==16*x+8*y+4*z,z)) show(solve(36==81*x+27*y+9*z,z)) p1=plot3d(-x-y+1,(x,0,1),(y,0,1),color='red') p2=plot3d(-4*x-2*y+9/4,(x,0,1),(y,0,1)) p3=plot3d(-9*x-3*y+4,(x,0,1),(y,0,1),color='green') (p1+p2+p3).show(viewer='tachyon') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{4}\right), b = \left(\frac{1}{2}\right), c = \left(\frac{1}{4}\right), d = 0, e = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, {\left(n + 1\right)}^{2} n^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -x - y + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -4 \, x - 2 \, y + \frac{9}{4}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -9 \, x - 3 \, y + 4\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[a = \left(\frac{1}{4}\right), b = \left(\frac{1}{2}\right), c = \left(\frac{1}{4}\right), d = 0, e = 0\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, {\left(n + 1\right)}^{2} n^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -x - y + 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -4 \, x - 2 \, y + \frac{9}{4}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[z = -9 \, x - 3 \, y + 4\right]