CSH4.2 MrG 2012.1018 Sequences

2967 days ago by CSH2012

#1a) {1,3,5,...,2*n-1} arithmetic: common difference=2 #1b) {2,4,8,...,2*2^(n-1)} geometric: common ratio=2 #1c) {81,27,9,3,1} = 81*(1/3)^(n-1), geometric: common ratio=1/3 show([2*n-1 for n in [1..5]]) show([2*2^(n-1) for n in [1..6]]) show([81*(1/3)^(n-1) for n in [1..5]]) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 3, 5, 7, 9\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[2, 4, 8, 16, 32, 64\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[81, 27, 9, 3, 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 3, 5, 7, 9\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[2, 4, 8, 16, 32, 64\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[81, 27, 9, 3, 1\right]
#2) {1/2,1/6,1/12,1/20,1/30,1/42}={1/(n^2+n)} [1/(n^2+n) for n in [1..6]] 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \frac{1}{30}, \frac{1}{42}\right]

#3)

a(n) = c + n*d

========

a(0) = c

a(1) = c + d

a(2) = c + 2d

a(3) = c + 3d

========

a(0) = c

a(3) = c + 3d

a(6) = c + 6d

a(9) = c + 9d

 

#4)

a(1) = c + d = 3

a(7) = c + 7d = 21

a(12) = c + 12d = ?

#4) a(12)=0+12*3=36 var('y') show(solve(x+y==3,y)) show(solve(x+7*y==21,y)) show(solve(-x+3==-x/7+3,x)) show(solve([x+y==3,x+7*y==21],(x,y))) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -x + 3\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -\frac{1}{7} \, x + 3\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = 0\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[x = 0, y = 3\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -x + 3\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[y = -\frac{1}{7} \, x + 3\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = 0\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[x = 0, y = 3\right]\right]
plot([-x+3,-x/7+3]) 
       
#5) {4,12,36,...,4*3^(n-1)} [4*3^(n-1) for n in [1..9]] 
       
\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]

#6)

a(1) = 1 = a*r^(0)

a(11) = 1024 = a*r^(10)

a(21) = (1)*2^(21-1)=2^20=1048576

a(n) = a*r^(n-1)

show(solve(1024==x^10,x)) show(2^20) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = 2 \, e^{\left(\frac{1}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{2}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{3}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{4}{5} i \, \pi\right)}, x = \left(-2\right), x = 2 \, e^{\left(-\frac{4}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{3}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{2}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{1}{5} i \, \pi\right)}, x = 2\right]
\newcommand{\Bold}[1]{\mathbf{#1}}1048576
\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = 2 \, e^{\left(\frac{1}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{2}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{3}{5} i \, \pi\right)}, x = 2 \, e^{\left(\frac{4}{5} i \, \pi\right)}, x = \left(-2\right), x = 2 \, e^{\left(-\frac{4}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{3}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{2}{5} i \, \pi\right)}, x = 2 \, e^{\left(-\frac{1}{5} i \, \pi\right)}, x = 2\right]
\newcommand{\Bold}[1]{\mathbf{#1}}1048576

 

#7)
a(n-1)= c+(n-1)*d
a(n)  = c+n*d
a(n+1)= c+(n+1)*d
(c+(n-1)*d + c+(n+1)*d)/2
(c+n*d-d + c+n*d+d)/2
(c+n*d + c+n*d)/2
(2*c+2*n*d)/2
c+n*d
#7)
a(n-1)= c+(n-1)*d
a(n)  = c+n*d
a(n+1)= c+(n+1)*d
(c+(n-1)*d + c+(n+1)*d)/2
(c+n*d-d + c+n*d+d)/2
(c+n*d + c+n*d)/2
(2*c+2*n*d)/2
c+n*d

 

var('c,n,d') ((c+(n-1)*d + c+(n+1)*d)/2).simplify_rational() 
       
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c
\newcommand{\Bold}[1]{\mathbf{#1}}d n + c

 

#8)
a(n-1) = a*r^(n-2)
a(n)   = a*r^(n-1)
a(n+1) = a*r^(n)
sqrt(a*r^(n-2)*a*r^(n))
sqrt(a^2*r^(2*n-2))
a*r^(n-1)

 

#8)

a(n-1)  = a*r^(n-2)

a(n)     = a*r^(n-1)

a(n+1) = a*r^(n)

var('a,r') (sqrt(a*r^(n-2)*a*r^(n))).simplify_radical() 
       
\newcommand{\Bold}[1]{\mathbf{#1}}r^{{\left(n - 1\right)}} {\left| a \right|}
\newcommand{\Bold}[1]{\mathbf{#1}}r^{{\left(n - 1\right)}} {\left| a \right|}
#9) {1,1,1,...} or {0,0,0,...} 
       
#10) {0,1/2,2/3,3/4,4/5,...,n/(n+1)} [(n/(n+1)).n() for n in range(10)] 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.000000000000000, 0.500000000000000, 0.666666666666667, 0.750000000000000, 0.800000000000000, 0.833333333333333, 0.857142857142857, 0.875000000000000, 0.888888888888889, 0.900000000000000\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, 0.900000000000000\right]
n=10 print "n\tn/(n+1)" while n<=100000: print n,"\t",(n/(n+1)).n() n=n*10 
       
n	n/(n+1)
10 	0.909090909090909
100 	0.990099009900990
1000 	0.999000999000999
10000 	0.999900009999000
100000 	0.999990000099999
n	n/(n+1)
10 	0.909090909090909
100 	0.990099009900990
1000 	0.999000999000999
10000 	0.999900009999000
100000 	0.999990000099999
#11) n=10 print "n\t(n-2)/(n-5)" while n<=100000: print n,"\t",((n-2)/(n-5)).n() n=n*10 
       
n	(n-2)/(n-5)
10 	1.60000000000000
100 	1.03157894736842
1000 	1.00301507537688
10000 	1.00030015007504
100000 	1.00003000150007
n	(n-2)/(n-5)
10 	1.60000000000000
100 	1.03157894736842
1000 	1.00301507537688
10000 	1.00030015007504
100000 	1.00003000150007
#12) n=10 print "n\tn/sqrt(n^2-1)" while n<=100000: print n,"\t",(n/sqrt(n^2-1)).n() n=n*10 
       
n	n/sqrt(n^2-1)
10 	1.00503781525921
100 	1.00005000375031
1000 	1.00000050000038
10000 	1.00000000500000
100000 	1.00000000005000
n	n/sqrt(n^2-1)
10 	1.00503781525921
100 	1.00005000375031
1000 	1.00000050000038
10000 	1.00000000500000
100000 	1.00000000005000