1203 MrG What's a Geometric Sequence

3002 days ago by MATH4R2013

#1) {2,6,18,54,...} a(n)=2*3^n show([a(n) for n in range(10)]) show(sum([a(n) for n in range(10)])) N=range(10) A=[a(n) for n in range(10)] AN=zip(N,A) plot(point(AN)) 
       


#2) {1,1/2,1/4,1/8,1/16,...}, a=1, r=.5 b(n)=1*(.5)^n show(sum([b(n) for n in range(5000)])) show(1*(1-.5^5000)/(1-.5)) 
       

#3) .99999... = .9+.09+.009+... .9/(1-.1) 
       
1.00000000000000
1.00000000000000
#4) .33333... = .3+.03+.003+... show(.3/(1-.1)) show((3/10)/(1-(1/10))) 
       

#5) .11111... = .1+.01+.001+... show(.1/(1-.1)) show((1/10)/(1-(1/10))) 
       

#72 p838: {1000,900,810,...} a=1000, r=.9 f(n)=1000*.9^n # takes 111 days to get a paycheck less than 1 cent! print([f(n) for n in range(111)]) # sum of all paychecks: show(1000*(1-.9^111)/(1-.9)) # sum of infinitely many paychecks? show(1000/(1-.9)) 
       
[1000, 900.000000000000, 810.000000000000, 729.000000000000,
656.100000000000, 590.490000000000, 531.441000000000, 478.296900000000,
430.467210000000, 387.420489000000, 348.678440100000, 313.810596090000,
282.429536481000, 254.186582832900, 228.767924549610, 205.891132094649,
185.302018885184, 166.771816996666, 150.094635296999, 135.085171767299,
121.576654590569, 109.418989131512, 98.4770902183612, 88.6293811965251,
79.7664430768726, 71.7897987691853, 64.6108188922668, 58.1497370030401,
52.3347633027361, 47.1012869724625, 42.3911582752162, 38.1520424476946,
34.3368382029252, 30.9031543826326, 27.8128389443694, 25.0315550499324,
22.5283995449392, 20.2755595904453, 18.2480036314007, 16.4232032682607,
14.7808829414346, 13.3027946472911, 11.9725151825620, 10.7752636643058,
9.69773729787525, 8.72796356808772, 7.85516721127895, 7.06965049015106,
6.36268544113595, 5.72641689702235, 5.15377520732012, 4.63839768658811,
4.17455791792930, 3.75710212613637, 3.38139191352273, 3.04325272217046,
2.73892744995341, 2.46503470495807, 2.21853123446226, 1.99667811101604,
1.79701029991443, 1.61730926992299, 1.45557834293069, 1.31002050863762,
1.17901845777386, 1.06111661199647, 0.955004950796827,
0.859504455717144, 0.773554010145430, 0.696198609130887,
0.626578748217798, 0.563920873396018, 0.507528786056416,
0.456775907450775, 0.411098316705697, 0.369988485035128,
0.332989636531615, 0.299690672878453, 0.269721605590608,
0.242749445031547, 0.218474500528393, 0.196627050475553,
0.176964345427998, 0.159267910885198, 0.143341119796678,
0.129007007817011, 0.116106307035309, 0.104495676331779,
0.0940461086986007, 0.0846414978287406, 0.0761773480458666,
0.0685596132412799, 0.0617036519171519, 0.0555332867254367,
0.0499799580528931, 0.0449819622476038, 0.0404837660228434,
0.0364353894205590, 0.0327918504785031, 0.0295126654306528,
0.0265613988875875, 0.0239052589988288, 0.0215147330989459,
0.0193632597890513, 0.0174269338101462, 0.0156842404291316,
0.0141158163862184, 0.0127042347475966, 0.0114338112728369,
0.0102904301455532, 0.00926138713099790]

[1000, 900.000000000000, 810.000000000000, 729.000000000000, 656.100000000000, 590.490000000000, 531.441000000000, 478.296900000000, 430.467210000000, 387.420489000000, 348.678440100000, 313.810596090000, 282.429536481000, 254.186582832900, 228.767924549610, 205.891132094649, 185.302018885184, 166.771816996666, 150.094635296999, 135.085171767299, 121.576654590569, 109.418989131512, 98.4770902183612, 88.6293811965251, 79.7664430768726, 71.7897987691853, 64.6108188922668, 58.1497370030401, 52.3347633027361, 47.1012869724625, 42.3911582752162, 38.1520424476946, 34.3368382029252, 30.9031543826326, 27.8128389443694, 25.0315550499324, 22.5283995449392, 20.2755595904453, 18.2480036314007, 16.4232032682607, 14.7808829414346, 13.3027946472911, 11.9725151825620, 10.7752636643058, 9.69773729787525, 8.72796356808772, 7.85516721127895, 7.06965049015106, 6.36268544113595, 5.72641689702235, 5.15377520732012, 4.63839768658811, 4.17455791792930, 3.75710212613637, 3.38139191352273, 3.04325272217046, 2.73892744995341, 2.46503470495807, 2.21853123446226, 1.99667811101604, 1.79701029991443, 1.61730926992299, 1.45557834293069, 1.31002050863762, 1.17901845777386, 1.06111661199647, 0.955004950796827, 0.859504455717144, 0.773554010145430, 0.696198609130887, 0.626578748217798, 0.563920873396018, 0.507528786056416, 0.456775907450775, 0.411098316705697, 0.369988485035128, 0.332989636531615, 0.299690672878453, 0.269721605590608, 0.242749445031547, 0.218474500528393, 0.196627050475553, 0.176964345427998, 0.159267910885198, 0.143341119796678, 0.129007007817011, 0.116106307035309, 0.104495676331779, 0.0940461086986007, 0.0846414978287406, 0.0761773480458666, 0.0685596132412799, 0.0617036519171519, 0.0555332867254367, 0.0499799580528931, 0.0449819622476038, 0.0404837660228434, 0.0364353894205590, 0.0327918504785031, 0.0295126654306528, 0.0265613988875875, 0.0239052589988288, 0.0215147330989459, 0.0193632597890513, 0.0174269338101462, 0.0156842404291316, 0.0141158163862184, 0.0127042347475966, 0.0114338112728369, 0.0102904301455532, 0.00926138713099790]