Hi, can somoene check why my solution and explain why i get the wrong answer.
the is to give an explicit formula (i. e., closed form) of a_{n} . a_{0}=2 , a_{1}=7 and a_{n}=5a_{n-1}-6a_{n-2}-2^{n-1} and n\geq 2 .
solution:
well to find out what a_{n}^{h} is not hard, i skip the solution for that and just write the answer. a_{n}^{h}= a3^{n}+b2^{n} . ok, so now we want to find out what a_{n}^{p} is. for that we "guess" the solution. well, the guess is that a_{n}^{p}=cn2^{n} . now lets plug a_{n}^{p} into a_{n}=5a_{n-1}-6a_{n-2}-2^{n-1} . we get after some work that -2cn+5cn-5c- 3cn+6c=-1 , so c=-1 .
a_{n}=a_{n}^{h}+a_{n}^{p}=a3^{n}+b2 ^{n}-1. so after we plug in a_{0}=2 and a_{1}=7 we get that a=2 and b=1.
therefore a_{n}=2*3^{n}+2^{n}-1 . but the answer is a_{n}=(n+1)2^{n}+3^{n} . i think my guess for a_{n}^{p} is wrong. write so i can see, and use the algoritm i am using and no other. in advance.