# Proof by induction

#### Randyyy

##### Junior Member
Hey, Mathhelp! Is my proof valid? I am new to induction so I am not too confident what I have done is correct.

Prove that 3n-1 is divisible by 2 $$\displaystyle \forall n \in\mathbb{Z}^+ ,n\geq 1$$
S(n)=3n-1=2k, $$\displaystyle k\in\mathbb{Z}$$
Base-case n=1: 31-1=2*1, true for n=1
S(p)=3p-1=2k
Assuming it is true for n=p, then it is true for n=p+1.
S(p+1)= 3p+1-1=2r, $$\displaystyle r\in\mathbb{Z}$$
S(p)=3(3p-1)=3*2k
S(p)=3p+1-3=6k
S(p)=3p+1-1=6k+2
S(p)=3p+1-1=2(3k+1)
(3k+1)=r and hence:
S(p)=3p+1-1=2r

#### pka

##### Elite Member
Prove that 3n-1 is divisible by 2 $$\displaystyle \forall n \in\mathbb{Z}^+ ,n\geq 1$$
Lets's suppose that $$3^p-1=2k$$ where $$\{p,k\}\subset\mathbb{Z}^+$$
Look at
\begin{align*}3^{p+1}-1&=3\cdot 3^p-1 \\&=3\cdot 3^p-3+2\\&=3\cdot (3^p-1)+2\\&=3\cdot (2k)+2\\&=2(3k+1)\text{ which is even} \end{align*}

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#### Randyyy

##### Junior Member
Lets's suppose that $$3^p-1=2k$$ where $$\{p,k\}\subset\mathbb{Z}^+$$
Look at
\begin{align*}3^{p+1}-1&=3\cdot 3^p-1 \\&=3\cdot 3^p-3+2\\&=3\cdot (3^p-1)+2\\&=3\cdot (2k)+2\\&=2(3k+1)\text{ which is even} \end{align*}
Thank you