Question

How many of the below given sequences is a solution to the following recurrence relation: a_n = a_n-1 + 2a_n-2 +2n - 9?

1.) a_n = 2ⁿ + n - 2

2.) a_n = 7 $LaTeX: \cdot$ 2ⁿ - n + 2

3.) a_n = 5(-1)ⁿ - n + 2

4.) a_n = 2 - n

5.) a_n = 2 $LaTeX: \cdot$ (-4)ⁿ + 3

Answer 1

a_n = a_n -1 + 2 a_n -2 + 2 n -9 (1) a_n = 2n + n -2 R.H: S = 2 (n -1) + (n -1) -2 + 2 [2 (n -2) + n -2 -2] + 2n -9 = 2n -2 + n -1 -2 + 4n -4 + 2n -8 + 2n -9 = 11n -26 notequalto a_n = L.H.S (2) a_n = 7.2n - n + 2 R.H.S = 7.2(n -1) - (n -1) + 2 + 2 [7.2 (n -2) - (n -2) + 2]+ 2n - n = 14n -14 - n + 2 + 28n -56 -2n + 8 + 2n -9 = 41n -68 notequalto a_n = L.H.S (3) a_n = 5(-1)_n - n + 2 = -6n + 2 R.H.S = -6(n -1)