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 [MATH]\forall n \in\mathbb{Z}^+ ,n\geq 1[/MATH]S(n)=3n-1=2k, [MATH]k\in\mathbb{Z}[/MATH]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, [MATH]r\in\mathbb{Z}[/MATH]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
Prove that 3n-1 is divisible by 2 [MATH]\forall n \in\mathbb{Z}^+ ,n\geq 1[/MATH]S(n)=3n-1=2k, [MATH]k\in\mathbb{Z}[/MATH]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, [MATH]r\in\mathbb{Z}[/MATH]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