Using mathematical induction, prove that n^3+2n is divisible by 3 for all integers n

To prove this we must use a neat mathematical technique called induction.

Induction works in the following way: If you show that the result being true for any integer implies it is true for the next, then you need only show that it is true for n=1 for it to be true for n=2 and then n=3 and so on.

Step 1: Show true for n=1

For n=1, n^3+2n=(1)^3+2(1)


3 is definitely divisible by 3 so the statement is true for n=1.

Step 2: Assume true for n=k

We assume that for any integer k, n^3+2n is divisible by 3. We can write this mathematically as:

k^3+2k=3m, where m is an integer

Step 3: Show true for k+1

For n=k+1,




Subbing in from part 2 for (k^3+2k), we get:



which is divisible by 3.


This means that the statement being true for n=k implies the statement is true for n=k+1, and as we have shown it to be true for n=1 the proof of the statement follows by induction.

James S. A Level Maths tutor, A Level Physics tutor, A Level Further ...

2 years ago

Answered by James, an A Level Further Mathematics tutor with MyTutor

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


£36 /hr

James G.

Degree: Mathematical Physics (Doctorate) - Nottingham University

Subjects offered:Further Mathematics , Physics+ 2 more

Further Mathematics

“Currently a 3rd year PhD student in Mathematical Physics. I'm very passionate about teaching as well as my subject area. Look forward to hearing from you.”

MyTutor guarantee

£26 /hr

Tadas T.

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered:Further Mathematics , Maths+ 3 more

Further Mathematics
-Personal Statements-
-Oxbridge Preparation-

“University of Oxford Maths and Philosophy student happy to help students learn and stay motivated!”

£20 /hr

Gwen W.

Degree: Chemistry and Physics (Bachelors) - St. Andrews University

Subjects offered:Further Mathematics , Maths

Further Mathematics

“Hi, I'm Gwen! I have a great interest and enthusiasm for maths and would love to use this to help you reach your full potential in exams.”

About the author

£20 /hr

James S.

Degree: Mathematics and Physics (MSci) (Masters) - Durham University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics

“First year Maths and Physics undergraduate at Durham University. Previous tutoring experience with A-level students”

MyTutor guarantee

You may also like...

Posts by James

A given star has a peak emission wavelength of 60nm, lies 7.10*10^19m away and the intensity of its electromagnetic radiation reaching the Earth is 3.33*10^-8Wm^-2. Calculate the star's diameter

Find the stationary point of y=3x^2-12x+29 and classify it as a maximum/minimum

Using mathematical induction, prove that n^3+2n is divisible by 3 for all integers n

Other A Level Further Mathematics questions

Prove that sum(k) from 0 to n is n(n+1)/2, by induction

Write 1 + √3i in modulus-argument form

Prove by induction that 11^n - 6 is divisible by 5 for all positive integer n.

Find the general solution for the determinant of a 3x3 martix. When does the inverse of this matrix not exist?

View A Level Further Mathematics tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss