CM 4.2 Sequences MrG 20111031

3676 days ago by calcpage123

#1) {1,7,13,19,...} = {1+(n-1)6} var('n') 1+(n-1)*6 
       
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, n - 5
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, n - 5
#2) {1/2,1/6,1/12,1/20,1/30,...} = {1/(n^2+n))} for i in range(1,7): show(1/(i^2+i)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{12}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{20}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{30}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{42}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{12}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{20}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{30}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{42}
for i in range(1,7): show((1/(i^2+i)).n()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}0.500000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}0.166666666666667
\newcommand{\Bold}[1]{\mathbf{#1}}0.0833333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.0500000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}0.0333333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.0238095238095238
\newcommand{\Bold}[1]{\mathbf{#1}}0.500000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}0.166666666666667
\newcommand{\Bold}[1]{\mathbf{#1}}0.0833333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.0500000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}0.0333333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.0238095238095238
#3)a(n)=c+(n-1)*d #3)b(n)=(c-d)+(n-1)*3d var('c,d') a(n)=c+(n-1)*d for i in range(0,10,3): show(a(i)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c - d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 2 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 5 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 8 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c - d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 2 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 5 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 8 \, d
b(n)=(c-d)+(n-1)*3*d for i in range(1,5): show(b(i)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c - d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 2 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 5 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 8 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c - d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 2 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 5 \, d
\newcommand{\Bold}[1]{\mathbf{#1}}c + 8 \, d
#4)a(1)=3, a(7)=21, a(12)=? #4)a(n) = c + (n-1)*d #4)a(n) = 3 + (n-1)*3 #4)a(12) = 3 + (12-1)*3 = 36 a(n)=3+(n-1)*3 for i in range(1,13): show(a(i)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}3
\newcommand{\Bold}[1]{\mathbf{#1}}6
\newcommand{\Bold}[1]{\mathbf{#1}}9
\newcommand{\Bold}[1]{\mathbf{#1}}12
\newcommand{\Bold}[1]{\mathbf{#1}}15
\newcommand{\Bold}[1]{\mathbf{#1}}18
\newcommand{\Bold}[1]{\mathbf{#1}}21
\newcommand{\Bold}[1]{\mathbf{#1}}24
\newcommand{\Bold}[1]{\mathbf{#1}}27
\newcommand{\Bold}[1]{\mathbf{#1}}30
\newcommand{\Bold}[1]{\mathbf{#1}}33
\newcommand{\Bold}[1]{\mathbf{#1}}36
\newcommand{\Bold}[1]{\mathbf{#1}}3
\newcommand{\Bold}[1]{\mathbf{#1}}6
\newcommand{\Bold}[1]{\mathbf{#1}}9
\newcommand{\Bold}[1]{\mathbf{#1}}12
\newcommand{\Bold}[1]{\mathbf{#1}}15
\newcommand{\Bold}[1]{\mathbf{#1}}18
\newcommand{\Bold}[1]{\mathbf{#1}}21
\newcommand{\Bold}[1]{\mathbf{#1}}24
\newcommand{\Bold}[1]{\mathbf{#1}}27
\newcommand{\Bold}[1]{\mathbf{#1}}30
\newcommand{\Bold}[1]{\mathbf{#1}}33
\newcommand{\Bold}[1]{\mathbf{#1}}36
#5)a(1)=4, a(2)=12, a(3)=36 #5)a(n)=c*r^(n-1) #5)a(n)=4*3^(n-1) a(n)=4*3^(n-1) for i in range(1,10): print(i,a(i)) 
       
(1, 4)
(2, 12)
(3, 36)
(4, 108)
(5, 324)
(6, 972)
(7, 2916)
(8, 8748)
(9, 26244)
(1, 4)
(2, 12)
(3, 36)
(4, 108)
(5, 324)
(6, 972)
(7, 2916)
(8, 8748)
(9, 26244)
[4*3^(i-1) for i in range(1,10)] 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[4, 12, 36, 108, 324, 972, 2916, 8748, 26244\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[4, 12, 36, 108, 324, 972, 2916, 8748, 26244\right]
def a(*n): return [4*3^(i-1) for i in n] a(2,4,6,8,9,3,4,6) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[12, 108, 972, 8748, 26244, 36, 108, 972\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[12, 108, 972, 8748, 26244, 36, 108, 972\right]
b(n)=4*3^(n-1) map(b,range(2,8,2)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[12, 108, 972\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[12, 108, 972\right]
#6)b(1)=1, b(11)=1024, b(21)=? #6)b(n)=c*r**(n-1) #6)b(n)=1*2**(n-1) b(n)=1*2**(n-1) [b(i) for i in range(1,22)] 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576\right]
#7) show(c+((n-1)-1)*d) show((c+(n-1)*d).simplify_rational()) show(c+((n+1)-1)*d) show(((c+((n-1)-1)*d+c+((n+1)-1)*d)/2).simplify_rational()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(n - 2\right)} d + c
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c - d
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c - d
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(n - 2\right)} d + c
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c - d
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c - d
#8) var('r') assume(c>0) show(c*r^((n-1)-1)) show(c*r^(n-1)) show(c*r^((n+1)-1)) show((sqrt(c*r^((n-1)-1)*c*r^((n+1)-1))).simplify_radical()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 2\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 1\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}r^{n} c
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 1\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 2\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 1\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}r^{n} c
\newcommand{\Bold}[1]{\mathbf{#1}}c r^{{\left(n - 1\right)}}
#9) {2,2,2,2,2,2,2,...} 
       
#10) [n/(n+1).n() for n in range(9)] 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.000000000000000, 0.500000000000000, 0.666666666666667, 0.750000000000000, 0.800000000000000, 0.833333333333333, 0.857142857142857, 0.875000000000000, 0.888888888888889\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.000000000000000, 0.500000000000000, 0.666666666666667, 0.750000000000000, 0.800000000000000, 0.833333333333333, 0.857142857142857, 0.875000000000000, 0.888888888888889\right]
for n in range(9): print(n/(n+1)) 
       
0
1/2
2/3
3/4
4/5
5/6
6/7
7/8
8/9
0
1/2
2/3
3/4
4/5
5/6
6/7
7/8
8/9