Proving the Sum of Squares of Odd Numbers: Algebraic Method

In summary, to prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4, you can use the formula (2n-1)^2 + (2n-1)^2 = 4(n^2 + n) + 2, where n is any integer. This formula can be derived by replacing the 1's in the original attempt with the general form of an odd number, 2n-1, and using the binomial expansion.
  • #1
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Homework Statement



Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.

Homework Equations



n/a

The Attempt at a Solution



not really sure what to do so excuse my bad attempt

where n is an integer:

(1^2 + 1^2)/4 = n + 2


thnx for you help

as per rules, please only give one step away at a time and explain please :D thnx
 
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  • #2
Can you write me a formula for the nth odd number?
 
  • #3
You need to replace those 1's with the general form of an odd number. Put all odd numbers in terms of all whole numbers and do the binomial expansion
 
  • #4
oops yeh i found out i should have done 2n-1 lol

silly me, soz
 

FAQ: Proving the Sum of Squares of Odd Numbers: Algebraic Method

1. What is a proof calculation?

A proof calculation is a mathematical process used to demonstrate the validity of a statement or theorem. It involves using logical reasoning and mathematical rules to show that the statement is true.

2. How do I solve a proof calculation?

To solve a proof calculation, you need to carefully read and understand the given statement or theorem. Then, use mathematical rules and logic to break it down into smaller, more manageable steps. Finally, use these steps to arrive at a conclusion that proves the statement to be true.

3. What are some common strategies for proof calculation?

Some common strategies for proof calculation include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. These strategies involve using logical reasoning and mathematical rules to show the truth of a statement.

4. How can I improve my skills in proof calculation?

To improve your skills in proof calculation, it is important to practice regularly and to familiarize yourself with different proof strategies. You can also seek help from a tutor or study group, and try to understand and learn from other people's proofs.

5. Are there any tips for writing a clear and concise proof?

Yes, there are a few tips for writing a clear and concise proof. First, make sure to clearly state your assumptions and theorems. Use precise and succinct language, and avoid unnecessary details. Additionally, break down the proof into smaller steps and clearly label each step. Finally, review and revise your proof to ensure it is logically sound and easy to follow.

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