1. discrete maths

I will let D be a function that contains elements n from the set of Natural Numbers: N

i define D(n) = 2 x 3^c
.... .. .................. c=0

(b) Prove this sentance by induction:
for all n which are elements from set Natural Numbers: N, D(n) = 3^(n+1) +1

Base case: when n is 1 D(1) = 3^(1+1) + 1 = 9 + 1 = 10

Inductive step: we assume that D(n) = 3^(n+1) is true for n = k, hence D(k) = 3^(k+1) + 1 is true
We prove that D(k) = 3^(k+1) + 1

sorry for my english i am from italy. this is what im quite unsure about what is happen here and if im doing it right, how do i prove now?

2. Re: discrete maths

I am very confused.

$\displaystyle GIVEN:\ n \in \mathbb N^+\ and\ D(n) = 2 * \sum_{c=0}^n3^c.$

$PROVE: D(n) = 3^{(n+1)} + 1.$

Is that correct?

$D(1) = 2(3^0 + 3^1) = 2(1 + 3) = 8 \ne 10 = 9 + 1 = 3^{(1+1)} + 1.$

3. Re: discrete maths

Originally Posted by JeffM
I am very confused.

$\displaystyle GIVEN:\ n \in \mathbb N^+\ and\ D(n) = 2 * \sum_{c=0}^n3^c.$

$PROVE: D(n) = 3^{(n+1)} + 1.$

Is that correct?

$D(1) = 2(3^0 + 3^1) = 2(1 + 3) = 8 \ne 10 = 9 + 1 = 3^{(1+1)} + 1.$
Oh my bad that is all correct except the statement that we have to prove by induction the end bit is -1 not +1.

So yeah all of its correct and i have to prove the sentance by induction:
Prove by induction: D(n)=3^(n+1)-1.

EDIT: The working out i done at top is that not correct because proof by induction it say you need base case and induction step.

4. Re: discrete maths

Can someone tell me how i prove it by induction please?

5. Re: discrete maths

Show true for $D(1)$ ... you can do this.

Assume true for $D(n)$, show true for $D(n+1)$

$\displaystyle D(n)=2\sum_{c=0}^n 3^c = 3^{n+1}-1$

$\displaystyle D(n+1) = 2\sum_{c=0}^{n+1} 3^c = 2\sum_{c=0}^n 3^c + (2 \cdot 3^{n+1}) = 3^{n+1}-1 + 2\cdot 3^{n+1}=$

$3 \cdot 3^{n+1}-1 = 3^{n+2}-1 =3^{(n+1)+1}-1$

in future, do not bump your thread ... thank you for cooperating.

6. Re: discrete maths

Originally Posted by PussyEnjoyer
hey bro thanks. That final answer 3^((n+1)+1) - 1, is that correct?
Yes, that is correct.

You sent me a private message about formatting. It may help others if I answer publicly.

This site supports a formatting system called LaTeX. Most of the regular tutors use it, and you can see how it works by hitting reply with quote instead of reply. Unless you are going to post regularly, I suggest that you concentrate on math rather than LaTeX because the latter is a fussy beast. If you are going to post frequently, you may as well learn it because your questions will be easier to understand. There is a forum here that will answer specific questions about LaTeX, and there are good teaching resources on the Internet.