2 are1, 2, 4, 8, ..., 2n,... and 1, -1, 1, -1, ..., (-1)n, ...
3. Consider the second-order linear homogeneous recurrence relation rk
= rk-1 + 2 rk-2.
a) The two sequences, bk and ck , which satisfy this
relation are found by solving the characteristic equation.
The characteristic equation of this relation is t2 - t - 2 = 0.
By factoring t2 - t - 2 = (t - 2) (t + 1), so the two values of t which satisfy the characteristic equation are t = 2 and t = -1.
Hence the two sequences which satisfy the recurrence
relation rk = rk-1 + 2 rk-2 for all integers k2 are
1, 2, 4, 8, ..., 2n,... and 1, -1, 1, -1, ..., (-1)n,
...
b) Now let an = 3 bn - cn for all n
0.
For all integers k2, the sequence
ak also satisfies the recurrence relation rk = rk-1 + 2 rk-2,
since
ak-1 + 2 ak-2 |
= | (3 bk+1 - ck+1) + 2 (3 bk-2 - ck-2) | by definition of an |
| = | 3 ( bk-1 + 2 bk-2 ) - (ck-1 + 2 ck-2 ) | by rearranging | |
| = | 3 bk - ck | since bk and ck satisfy the original relation rk = rk-1 + 2 rk-2 | |
| = | ak | by definition of an |