Was given this as a question.
"Use iteration to guess an explicit formula for the recursively defined sequence and then prove that the formula is a solution to the recurrence using induction:
$a_{k} = ka_{k-1}$, for all integers k>1 $a_{0} = 1$"
(English: a sub k equals k times a sub k minus one, a sub zero equals 1)
The problem with this question is I cannot determine a ratio as k is equal to the index of the sequence. Given this equation I get the sequence:
1,1,2,6,24,120
It is obvious to me that I need the result of the previous index and multiply it by the current one. I tried using geometric and arithmetic sequencing methods, however, the partial sum of a geometric sequence doesn't account for the +1 increase in what otherwise is a constant ratio. I cannot find any such instance of this question anywhere and I believe the question is incomplete, but if I have the right info, where do I go?
Best Answer
The induction hypothesis is $a_n = \prod\limits_{k=1}^nk.$ This is true for the base case as $a_0 = 1$.
Now $a_{n+1} = (n+1)a_n = (n+1)\prod\limits_{k=1}^nk = \prod\limits_{k=1}^{n+1}k$, which proves the hypothesis.