SOLUTION: Prove algebraically that (2n+1)^2-(2n+1) is an even number for all positive integer values of n. Please help :)

Algebra.Com
Question 924485: Prove algebraically that
(2n+1)^2-(2n+1) is an even number
for all positive integer values of n.
Please help :)
Found 4 solutions by Fombitz, amalm06, ikleyn, MathTherapy:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!


So then, for any value ,
is even.
is odd.
The product of an even number and an odd number is always even.


Answer by amalm06(224)   (Show Source): You can put this solution on YOUR website!
Suppose it's an odd number instead. That is,

=, for some k∈Z

On the other hand,

, which is an even number. This contradicts the original hypothesis.


Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
It is easy to prove more general statement: 

    For any integer n the number    is even.


    Indeed,   = n*(n-1)  is the product of two consecutive integers.

    Of the two, one integer inevitably is even.


    Therefore, the product,  n*(n-1)  is even.


    Hence, the original    is even.


Answer by MathTherapy(10551)   (Show Source): You can put this solution on YOUR website!

Prove algebraically that
(2n+1)^2-(2n+1) is an even number
for all positive integer values of n.
Please help :)


(2n + 1)[(2n + 1) – 1]

(2n + 1)(2n + 1 – 1)

(2n + 1)(2n), or 2n(2n + 1)

2n is an EVEN NUMBER, and adding 1 to an even number creates an ODD number

Therefore,  

RELATED QUESTIONS


Prove that (2n+1)^2 - (2n-1)^2 is a multiple of 8 for all positive integer values of n.  (answered by ikleyn)

Let n be a positive integer.  Then 2n is always an even number. We say that 2n is the... (answered by richard1234)

use mathematical induction to prove that 
1^2 + 2^2 + 3^2 +...+ n^2 = n(n+1)(2n+1)/6... (answered by solver91311)

Prove by induction that for all positive integers value of n:
{{{5^(2n)+3n-1}}} is an... (answered by Edwin McCravy)

Prove by mathematical induction,1^2+3^2+....(2n+1)^2=((n+1)(2n+1)(2n+3))/3 where 'n' is a  (answered by ikleyn)

Prove by mathematical induction that:


2^2n - 1 is divisible by 3 for all positive... (answered by Edwin McCravy)

Prove by mathematical induction that 3^(2n)-8n-1, n is a positive integer, is a multiple... (answered by Edwin McCravy)

 Prove that 2C(2n − 1,n) = C(2n,n) for all n >... (answered by greenestamps)

prove that the product of any three consecutive numbers is even. 
So far I have tried ,   (answered by josgarithmetic,tenkun,amalm06,ikleyn)