Problem set with 5 proofs involving odd, even, parity, etc.

[deme76]

Homework Statement

1.Prove that if a and b are both odd, then a^2 b^2 is also odd.
2.Two integers are not the same parity if they are both even or both odd.
Prove that if x and y are of the same parity, then x+y is even.
3.Prove that if m-5 is odd, then (m-5)^(2 ) is odd.
4.Show that ∛2 is an irrational number.
5.Prove by induction that 1^2+ 2^2+⋯+ n^2= 1/6 (n)(n+1)(2n+1)

Relevant Equations

Show that ∛2 is an irrational number.
Assume ∛(2 ) rational
we can therefore say ∛2
= a⁄(b ) where a ,b are integers,and a and b are coprime
2= a^3/b^3
2b^3= a^3
hence,a is an even integers
we can say ,a=2n where m is an integer
〖2b〗^(3 )= (2m)^3
2b^3=8m^3
b^3= 〖4m〗^3
so b is also even.This complete the contradiction where we assumed
a and b were coprime.
Therefore, ∛2 is an irrational number
5.Prove by induction that 1^2+ 2^2+⋯+ n^2= 1/6 (n)(n+1)(2n+1)

help me in my problem set
qs

 

[PeterDonis]

help me in my problem set

You need to show some attempt at a solution. What have you tried?

 

[PeterDonis]

Moderator's note: Thread moved to precalculus math homework.

 

[SammyS]

You need to show some attempt at a solution. What have you tried?

All his/her working is in the Relevant Equations section.

 

[Hall]

Homework Statement:: 1.Prove that if a and b are both odd, then a^2 b^2 is also odd.
2.Two integers are not the same parity if they are both even or both odd.
Prove that if x and y are of the same parity, then x+y is even.
3.Prove that if m-5 is odd, then (m-5)^(2 ) is odd.
4.Show that ∛2 is an irrational number.
5.Prove by induction that 1^2+ 2^2+⋯+ n^2= 1/6 (n)(n+1)(2n+1)
Relevant Equations:: Show that ∛2 is an irrational number.
Assume ∛(2 ) rational
we can therefore say ∛2
= a⁄(b ) where a ,b are integers,and a and b are coprime
2= a^3/b^3
2b^3= a^3
hence,a is an even integers
we can say ,a=2n where m is an integer
〖2b〗^(3 )= (2m)^3
2b^3=8m^3
b^3= 〖4m〗^3
so b is also even.This complete the contradiction where we assumed
a and b were coprime.
Therefore, ∛2 is an irrational number
5.Prove by induction that 1^2+ 2^2+⋯+ n^2= 1/6 (n)(n+1)(2n+1)

help me in my problem set
qs

I reckon that it would be better if you provide a lemma that if $a^3$ is even, then $a$ is even, and then go for proving that cube root 2 is irrational. The reason for that is, it is standard to assume $a$ to be even when $a^2$ is given to be even, but the case of cube is not standard, so, we should prove it first.