Prove that the square of an odd number is always 1 more than a multiple of 4

First, we need to define what an odd number is. An odd number can be expressed as the product of 2n+1, where n can be any whole number bigger than 0. You can check this by inputting a few numbers for n (say 1,2,3) and making sure they're odd.Now, to find the square of any odd number, we square the expression we just came up with - remembering to fully multiply out our brackets (i.e multiply all the individual components together, and add the like terms)(2n+1)^2= (2n+1)(2n+1)= 4n^2 + 4n + 1Let's have another look at the question - it originally asked us to prove that an odd number is one bigger than a multiple of 4. Now if we look at the expression we came up with, we notice that the first two terms can be factorised into a simpler expression, as they both share 4 and n, as a factor. Here, we're only trying to prove it's bigger than a multiple of 4. Therefore, we can simply express (2n+1)^2= 4(n^2 + n) + 1So for any n, the product will be muliplied by 4, then plus 1.

Answered by Emanuel D. Maths tutor

3387 Views

See similar Maths GCSE tutors

Related Maths GCSE answers

All answers ▸

Solve the simultaneous equation 2x + y = 18 and x - y = 6


Two shops have deals for purchasing pens: "3 for £2" and "5 for £3" . Mr. Papadopoulos wants to buy 30 pens for his class in school, which deal should he use if he wants to spend the least amount of money possible, and how much will he spend?


How do you solve the following simultaneous equations? 4x-3y=18, 7x+5y=52


What is the solution to x^2 + 5x - 7 = 0


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences