CSH 4.3 MrG 2012.1023 Series

2971 days ago by CSH2012

#1&2)

a + (a+d) + (a+2d) + ... + (a+(n-2)d) + (a+(n-1)d) + (a+nd) = (n+1)(a + a + nd)/2

#3) sum(1/(n*(n+1))=1-1/(n+1) 1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6) 
       
1/1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6 
       

#4&5)
S=c+cr+cr^2+....+cr^(n-1)
rS=  cr+cr^2+....+cr^(n-1)+cr^n

S-rS=c-cr^n
S(1-r)=C(1-r^n)
S=(c(1-r)^n)/(1-r)

#6)sigma(d*10^d,d,1,6) def sigma(n): sum=0 for d in [1..n]: sum += d*10^d return sum sigma(6) 
       
#7)sigma(2^k-k,k,1,n) def sigma2(n): sum=0 for k in [1..n]: sum += 2^k-k return sum sigma2(30) 
       
#8a) show: 1+2+3+...+n = n(n+1)/2 #8a) let: 1+2+3+...+n = a*n^2+b*n+c #8a) if n=0: 0 == c #8a) if n=1: 1 == a+b+c #8a) 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]) 
       




#8b) show: 1^2+2^2+3^2+...+n^2 = n(n+1)(2n+1)/6 #8b) let: 1^2+2^2+3^2+...+n^2 = a*n^3+b*n^2+c*n+d #8b) if n=0: 0==d #8b) if n=1: 1==a+b+c+d #8b) if n=2: 5==8*a+4*b+2*c+d #8b) 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='jmol') 
       




#8c) show: 1^3+2^3+3^3+...+n^3 = (n(n+1)/2)^2 #8c) let: 1^3+2^3+3^3+...+n^3 = a*n^4+b*n^3+c*n^2+d*n+e #8c) if n=0: 0==e #8c) if n=1: 1==a+b+c+d+e #8c) if n=2: 9==16*a+8*b+4*c+2*d+e #8c) if n=3: 36==81*a+27*b+9*c+3*d+e #8c) 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') 
       




show(solve([x+y-z==-1,4*x-3*y+2*z==16,2*x-2*y-3*z==5],x,y,z)) show(solve(x+y-z==-1,z)) show(solve(4*x-3*y+2*z==16,z)) show(solve(2*x-2*y-3*z==5,z)) p1=plot3d(x+y+1,(x,-4,4),(y,-4,4),color='red') p2=plot3d(-2*x+3*y/2+8,(x,-4,4),(y,-4,4)) p3=plot3d(2*x/3-2*y-5/3,(x,-4,4),(y,-4,4),color='green') (p1+p2+p3).show(viewer='tachyon') 
       



show(solve([x-y+z==-4,x+y+z==6,9*x+3*y+z==0],x,y,z)) show(solve(x-y+z==-4,z)) show(solve(x+y+z==6,z)) show(solve(9*x+3*y+z==0,z)) p1=plot3d(-x+y-4,(x,-10,10),(y,-10,10),color='red') p2=plot3d(-x-y+6,(x,-10,10),(y,-10,10)) p3=plot3d(-9*x-3*y,(x,-10,10),(y,-10,10),color='green') (p1+p2+p3).show(viewer='tachyon')