A sequence is recursively defined by, t1=1, t2=2, t2n=2t2n-2, and t2n+1=3t2n-1
Is there a question in there somewhere?
This is a triangular number sequence and algebra is required.
To find the sequence, we can start by listing out the terms according to the given recursion formula and initial terms:
t1 = 1
t2 = 2
t3 = 2 * t1 = 2 * 1 = 2
t4 = 2 * t2 = 2 * 2 = 4
t5 = 3 * t3 = 3 * 2 = 6
t6 = 2 * t4 = 2 * 4 = 8
t7 = 3 * t5 = 3 * 6 = 18
...
We can observe that for even indices (2n), the term is double the term at index (2n - 2), and for odd indices (2n + 1), the term is three times the term at index (2n - 1).
Using this information, we can continue generating the sequence:
t8 = 2 * t6 = 2 * 8 = 16
t9 = 3 * t7 = 3 * 18 = 54
t10 = 2 * t8 = 2 * 16 = 32
t11 = 3 * t9 = 3 * 54 = 162
t12 = 2 * t10 = 2 * 32 = 64
t13 = 3 * t11 = 3 * 162 = 486
...
We can see that for even indices, the terms keep doubling, and for odd indices, the terms keep tripling.
So, the sequence generated by the given recursion formula is as follows:
1, 2, 2, 4, 6, 8, 18, 16, 54, 32, 162, 64, 486, ...