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

Question 1185342: Prove that (2n+1)^2 - (2n-1)^2 is a multiple of 8 for all positive integer values of n.
Answer by ikleyn(52798) (Show Source): You can put this solution on YOUR website!
.
Prove that (2n+1)^2 - (2n-1)^2 is a multiple of 8 for all positive integer values of n.
~~~~~~~~~~~~~~~~

    (2n+1)^2 - (2n-1)^2 = (4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n,  which is a MULTIPLE of 8 for any integer n.

Proved, solved and explained.

And completed.

RELATED QUESTIONS
Prove algebraically that (2n+1)^2-(2n+1) is an even number for all positive integer... (answered by Fombitz,amalm06,ikleyn,MathTherapy)

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

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

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

Use the method of mathematical induction to establish the following 1.... (answered by ikleyn)

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 that if n is a positive integers of 7^2n is an integer positive of... (answered by Alan3354)

please help me with this use the method of mathematical induction to prove the following (answered by math-vortex)

Prove by mathematical induction that: 2^2n - 1 is divisible by 3 for all positive... (answered by Edwin McCravy)