Let {an} be the (unique) sequence satisfying the difference equation: an+1 = (an + an-1)/2 where a0 = 0, a1=1

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Let {an} be the (unique) sequence satisfying the difference equation: an+1 = (an + an-1)/2 where a0 = 0, a1=1

[From: ] [author: ] [Date: 11-11-22] [Hit: ]

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Find limn→∞ an.

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a0 = 0
a1 = 1
a2 = (1 + 0)/2 = 1/2
a3 = (1+1/2)/2 = 3/4
a4 = (3/4+1/2)/2 = 5/8
a5 =(5/8+3/4)/2 = 11/16
an+1 = (an+(an-2+an-1)/2)/2 as n-----> infinity = 2/3

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The solution of this difference equation is:
a[n] = (2/3) -(2/3)(-1/2)^n
And therefore, lim(n→∞) a[n] = 2/3

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I dunno O:

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LOL

1

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