#1) a(n)=a(n-1)+a(n-2)
#1) a(0)=1
#1) a(1)=1
#1) 1,1,2,3,5,8,13,21,....
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#2)
def f(n):
return 1/(n*(n+1))
f(4)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{20}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{20}
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#3)a(0)=c, a(1)=c+d, a(2)=c+2d, a(3)=c+3d, ...
#3)a(0)=c, a(3)=c+3d, a(6)=c+6d, a(9)=c+9d, ...
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#4)a(1)=3=c+d, a(7)=21=c+7d, a(12)=c+12d=36
var('c,d')
solve((3==c+d,21==c+7*d),(c,d))
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[c = 0, d = 3\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[c = 0, d = 3\right]\right]
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plot((-x+3,-x/7+3),-10,10,aspect_ratio=1)
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#5)
def seq(n):
return 4*3**n
seq(0)
\newcommand{\Bold}[1]{\mathbf{#1}}4
\newcommand{\Bold}[1]{\mathbf{#1}}4
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for i in range(10):
print seq(i)
4 12 36 108 324 972 2916 8748 26244 78732 4 12 36 108 324 972 2916 8748 26244 78732 |
#6)a(0)=1=a*r**0, a=1
#6)a(10)=1024=1*r**10, r=2
#6)a(n)=1*2**n
def geo(a,r,n):
return a*r**n
geo(1,2,20)
\newcommand{\Bold}[1]{\mathbf{#1}}1048576
\newcommand{\Bold}[1]{\mathbf{#1}}1048576
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for i in range(21):
print(geo(1,2,i))
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 |
#7)
var('c,d,n')
term1=c+(n-1)*d
term2=c+(n+1)*d
sum=expand(term1+term2)
avg=sum/2
show(term1)
show(term2)
show(sum)
show(avg)
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#8)
var('a,r,n')
assume(a>0)
term1=a*r**(n-1)
term2=a*r**(n+1)
prod=simplify(term1*term2)
avg=(sqrt(prod)).simplify_radical()
show(term1)
show(term2)
show(prod)
show(avg)
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#9)
def silly(n):
return 11
for i in range(10):
print silly(i)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 |
#10)
def weird(n):
return n/(n+1)
for i in range(10):
print weird(i).n()
0.000000000000000 0.500000000000000 0.666666666666667 0.750000000000000 0.800000000000000 0.833333333333333 0.857142857142857 0.875000000000000 0.888888888888889 0.900000000000000 0.000000000000000 0.500000000000000 0.666666666666667 0.750000000000000 0.800000000000000 0.833333333333333 0.857142857142857 0.875000000000000 0.888888888888889 0.900000000000000 |
#11)
def weird2(n):
return (n-2)/(n-5)
for i in [6..600000]:
if i%60000==0:
print i,weird2(i).n()
60000 1.00005000416701 120000 1.00002500104171 180000 1.00001666712964 240000 1.00001250026042 300000 1.00001000016667 360000 1.00000833344908 420000 1.00000714294218 480000 1.00000625006510 540000 1.00000555560700 600000 1.00000500004167 60000 1.00005000416701 120000 1.00002500104171 180000 1.00001666712964 240000 1.00001250026042 300000 1.00001000016667 360000 1.00000833344908 420000 1.00000714294218 480000 1.00000625006510 540000 1.00000555560700 600000 1.00000500004167 |
#12)
def weird2(n):
return n/sqrt(n**2-1)
for i in [10..100]:
if i%10==0:
print i,weird2(i).n()
10 1.00503781525921 20 1.00125234864352 30 1.00055601894761 40 1.00031264656071 50 1.00020006002001 60 1.00013891783077 70 1.00010205643748 80 1.00007813415647 90 1.00006173411124 100 1.00005000375031 10 1.00503781525921 20 1.00125234864352 30 1.00055601894761 40 1.00031264656071 50 1.00020006002001 60 1.00013891783077 70 1.00010205643748 80 1.00007813415647 90 1.00006173411124 100 1.00005000375031 |
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