How do you prove induction?

The inductive step in a proof by induction is to show that for any choice of k, if P(k) is true, then P(k+1) is true. Typically, you’d prove this by assum- ing P(k) and then proving P(k+1). We recommend specifically writing out both what the as- sumption P(k) means and what you’re going to prove when you show P(k+1).

What is the divisibility rule for 9?

The rule for divisibility by 9 is similar to divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9. Example: Consider 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9.

How to prove $3n(n+1)$ is divisible by 6?

To stay within the spirit of the problem, the fact that $3n(n+1)$ is divisible by 6 should also be proved by induction.$\\endgroup$ – mathguy Aug 1 ’16 at 13:33 Add a comment | 2 $\\begingroup$ No need for induction. $$n^3-n=n(n^2-1)=n(n-1)(n+1)$$ which are three consecutive integers. So one must be divisible by 3.

What is the next step in mathematical induction?

The next step in mathematical induction is to go to the next element after k and show that to be true, too: If you can do that, you have used mathematical induction to prove that the property P is true for any element, and therefore every element, in the infinite set.

How do you prove that (1 + 4) = 1?

First prove the base case n = 1. Then induct and make use of the fact that to conclude what you want. Of course you would still need induction or something to prove this identity. the only term in ( 1 + 4) n not being multiplied by a power of 4 is 1 but it disappears due to the − 1.

Is the expression 9^N+1 – 4^N + 5(4^n) divisible by 5?

Step 1 – n =1 we have 9–4 = 5 so the result is true for n=1. 9^n+1 – 4^n+1 = 9 (9^n) – 9 (4^n) + 5 (4^n) = 9 ( 9^n – 4^n) – 5 (4^n). So it should be obivous that the expression is divisible by 5 since 5 (4^n) is divisble by 5 and we have our old friend (9^n – 4^n) which is divisible by five, so the entire expression is divisible by 5.