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Carolyn Beck 2022-06-29 Answered
I want to solve the equation
b n = b n 1 + 2 b n 2 + n + 1
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Answers (1)

pyphekam
Answered 2022-06-30 Author has 27 answers
b n = b n 1 + 2 b n 2 + n + 1
The guess is
b n = c n + d
c n + d = c ( n 1 ) + d + 2 ( c ( n 2 ) + d ) + n + 1
c + 2 c n 4 c + 2 d + n + 1 = 0
5 c + 1 + 2 d + n ( 2 c + 1 ) = 0
2 c + 1 = 0 c = 1 2
5 2 + 1 + 2 d = 0 d = 7 4
So that:
b n = c 1 ( 1 ) n + c 2 2 n n 2 7 4
For b 1 = b 0 = 1
b 0 = c 1 + c 2 7 4
c 1 + c 2 = 11 4
Then
b 1 = c 1 + 2 c 2 1 2 7 4
c 1 + 2 c 2 = 13 4
Then we have:
c 1 = 3 4 , c 2 = 2
b n = c 1 ( 1 ) n + c 2 ( 2 n ) n 2 7 4
b n = 3 4 ( 1 ) n + 2 ( 2 n ) n 2 7 4
b 2 = 3 4 + 2 ( 2 2 ) 2 2 7 4
b 2 = 6
As expected.
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