#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]])
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#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]
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#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)))
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plot([-x+3,-x/7+3])
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#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]
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#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)
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#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
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#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|}
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#9) {1,1,1,...} or {0,0,0,...}
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#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]
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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 |
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