1359 views

### Prove that the difference of any two consecutive square numbers is odd

It is important we first define what we mean by an odd and even number.
An even number is any integer (whole number) number divisible by 2 so we can express any even number as 2x where x is any integer. When counting, every even number is followed by an odd number; 1,2,3... etc.
We can then express any odd number as 2x+1 as it will just be the next number after 2x i.e. add one.
Now any square number can be expressed as n^2 where n is any integer. The next square number can also be written as (n+1)^2 since it will be the square of the next number after n i.e. n+1.
As such, the difference of any two consecutive square numbers can be written as (n+1)^2 - n^2
Expanding this we get (n^2 + 2n + 1) - n^2
This reduces to 2n+1 since the n^2 values cancel.
Since any odd number can be written in the form 2x+1  where x is any integer as earlier defined, 2n+1 is an odd number for any value of n which completes the proof.

12 months ago

Answered by Amar, who has applied to tutor GCSE Maths with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 531 SUBJECT SPECIALISTS

£18 /hr

Luke B.

Degree: Mathematics (Masters) - Sheffield University

Subjects offered:Maths, Further Mathematics + 3 more

Maths
Further Mathematics
.STEP.
.MAT.
-Personal Statements-

“I am a fun, engaging and qualified tutor. I'd love to help you with whatever you need, giving you the support you need to be the best you can be!”

£24 /hr

Sioned D.

Degree: Law (Bachelors) - University College London University

Subjects offered:Maths, History+ 4 more

Maths
History
English
Biology
.LNAT.
-Personal Statements-

“Hello! I'm a University College London Final year Law Student with a great deal of tutoring experience in English, Maths, Sciences and Entrance Exams. ”

£22 /hr

Sam F.

Degree: Economics with Placement (Bachelors) - Bath University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Extended Project Qualification
Economics

“Studying for BSc Hons Economics, A level economics, maths and physics. Able to tutor GCSE/AS/A2 Economics, Maths and GCSE physics! ”

£18 /hr

Amar H.

Degree: Mathematics (Bachelors) - Manchester University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
-Personal Statements-

“About Me: Hi, my name is Amar and I'm currently reading Maths at the University of Manchester. I have a real passion for the subject and love teaching it to others and hope my students will love it as much as I do.  Having being tutor...”

MyTutor guarantee

### You may also like...

#### Other GCSE Maths questions

N = 2A + B. A is a two-digit square number. B is a two-digit cube number. What is the smallest possible value of N?

Each day Chen buys pasta or a sandwich. Where Pasta costs £3.20 and Sandwich costs £2.75. For 45 days, the relative frequency of Chen buying a sandwich was one fifth. Altogether, how much did he pay?