Recurrence Relation 系列

Both problems have to do with the geometric series
圖片參考：https://s.yimg.com/lo/api/res/1.2/pDHj3gP3ZMtwotlE3Qg4iw--/YXBwaWQ9dHdhbnN3ZXJzO3E9ODU-/http://i187.photobucket.com/albums/x22/cshung/7015031700091-2_zps7n0rnv1a.png


2015-03-18 08:57:52 補充：
I knew that solution based on characteristic equation.

That feel like reciting answer, the method doesn't tell you 'why' that works.

Thanks - I will try LyX :) Loved the typesetting you used in the other questions.

When it is known that b = -5 and c = 6,

that is, a[n + 2] - 5 a[n + 1] + 6a[n] = 0.

Consider the characteristic equation

x² - 5x + 6 = 0

(x - 2)(x - 3) = 0

x = 2 or x = 3

Therefore, a[n] = A(2^n) + B(3^n)

a[1] = 1 gives 2A + 3B = 1
a[2] = 0 gives 4A + 9B = 0

2015-03-18 04:36:54 補充：
A = 3/2 and B = -2/3.

Thus,

a[n] = (3/2)(2^n) - (2/3)(3^n)

a[n] = 3 [2^(n - 1)] - 2[3^(n - 1)]

2015-03-18 04:39:31 補充：
For LaTeX typesetting, if you use section* rather than section, you can get rid of the numbers 1, 2.

Also you can try using LyX if you are interested, that is much more convenient in editing LaTeX document comparing to editing codes directly.