Use mathematical induction to prove the statement is true for all positive integers n.

The integer n3 + 2n is divisible by 3 for every positive integer n

Question

Use mathematical induction to prove the statement is true for all positive integers n.

The integer n3 + 2n is divisible by 3 for every positive integer n

anonymous · Answer

i take it it is $n^3+2n$ right?

anonymous · Answer

im actually really confuded

anonymous · Answer

k, have you ever done a proof by induction?

anonymous · Answer

nope:(

anonymous · Answer

damn
then we have  a long way to go
there is a method, and a bit of algebra will be used
step one (the easy step)
show that it is true if $n=1$ i .e. replace $n$ by $1$ and see what you get

anonymous · Answer

what do you get if $n=1$ ?

anonymous · Answer

3?

anonymous · Answer

yes
is 3 divisible by 3?

anonymous · Answer

yes?

anonymous · Answer

lol, yes!
so it is true if $n=1$ and the first step is done

anonymous · Answer

step two, assume it is true if $n=k$ 
this may be confusing , but just replace $n$ by $k$ and let me know what you get

anonymous · Answer

wait so like  k3 + 2k

anonymous · Answer

yes, $k^3+2k$ exactly
so we assume that $k^2+2k$ is divisible by $3$

anonymous · Answer

with that assumption, (now comes the only hard part) we want to show that it is true if $n=k+1$
replace $n$ by $k+1$and let me know what you get

anonymous · Answer

k+1=k+1?

anonymous · Answer

replace $n$ in the expression $n^3+2n$ by $k+1$

anonymous · Answer

okay k+1^3+2(k+1)

anonymous · Answer

right
now comes the algebra

anonymous · Answer

can you expand this nonsense 
$$(k+1)^3+2(k+1)$$?

anonymous · Answer

(k+1)(k+1)(k+1)+2k+2?

anonymous · Answer

yeah, that is step one

anonymous · Answer

i don't mean to torture you 
$$(k+1)^2=k^3+3k^2+3k+1$$

anonymous · Answer

so you have 
$$k^3+3k^2+3k+1+2k+2$$

anonymous · Answer

oh no scratch that
lets take a $k^3+2k$ out of this and write it as 
$$k^3+2k+3k^2+3k+3$$

anonymous · Answer

or better yet 
$$\color{red}{k^3+2k}+\color{blue}{3(k^2+k+1)}$$

anonymous · Answer

we know that $\color{red}{k^3+2k}$ is divisible by 3 "by induction" i.e. it is what we assumed to be true at the beginning of step 2
and we know that $\color{blue}{3(k^2+k+1)}$ is divisible by 3 because of that 3 out front
therefore the whole thing has to be divisible by 3, and that is what you had to prove

anonymous · Answer

WOW THANK YOU SO MUCH ♥