In this sequence, tk is a factorial number, often written k!. Show that tk=k!=k(k-1)(k-2)...(3)(2)(1)
I don't get it
k!=k(k-1)(k-2)...(3)(2)(1) by definition,
what's to "show" ?
what did they speak
language
To show that tk = k! = k(k-1)(k-2)...(3)(2)(1), we can use mathematical induction.
Step 1: Base case
Let's begin by checking the base case, which is k = 1.
When k = 1, t1 = 1! (by definition of the sequence) = 1 (by the definition of factorial).
So the base case holds true.
Step 2: Inductive hypothesis
Assume that for some positive integer n, tn = n! = n(n-1)(n-2)...(3)(2)(1). This is our inductive hypothesis. We want to show that this holds true for n+1 as well.
Step 3: Inductive step
Now we will show that if the hypothesis is true for n, then it is also true for n+1.
When k = n+1, we have:
tn+1 = (n+1)! (by definition of the sequence)
= (n+1)(n)(n-1)(n-2)...(3)(2)(1) (by the definition of factorial, which states that (n+1)! = (n+1)(n)!)
By the inductive hypothesis, we know that tn = n! = n(n-1)(n-2)...(3)(2)(1).
Multiplying tn by (n+1) gives us:
(n+1) * [n(n-1)(n-2)...(3)(2)(1)] = (n+1)(n)(n-1)(n-2)...(3)(2)(1)
Comparing the expression above with tn+1, we see that they are equal. Hence, the inductive hypothesis holds true for n+1 as well.
Therefore, by mathematical induction, we have shown that for any positive integer k, tk = k! = k(k-1)(k-2)...(3)(2)(1).