MYTUTOR SUBJECT ANSWERS

384 views

Prove by induction that n^3+5n is divisible by 3 for every natural number.

Proof by induction has three core elements to it. To start with you must prove that the statement is true for the 'basic case'. For the most part this is 1, but some questions state it is higher.

Do this by subbing 1 into the equation and ensuring that it is divisible by 3 

1^3 =1

5(1)=5 

1+5=6 6/3=2 Therefore divisible by three and true for 1.

Then in order to futher prove it, we are going to assume that this is true for n=k

leaving us with the equation k^3+5k=3a as it is divisible by 3.

The next stage is to prove true for n=k+1.

Do this by subbing k+1 into the original equation:

(k+1)^3 +5(k+1)

multiplying this out gives:

k^3+3k^2+3k+1+5k+5

Now we have already established that k^3+5k=3a so through rearranging, k^3=3a-5k.

Subbing this into the k+1 equation gives us:

3d+3k^2+3k-6. Each element is a multiple of three so by taking three out leaves us:

3(d+k^2+k-2) which is a multiple of three and thus divisible by three.

Then leave a concluding statement along the lines of:

'As n^3+5n is true for n=k, then it is true for n=k+1. As it is true for n=1, then it must be true for n is greater than 1'

Philip D. A Level Maths tutor, 13 Plus  Maths tutor, 11 Plus Maths tu...

9 months ago

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


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

120 SUBJECT SPECIALISTS

£20 /hr

Luke B.

Degree: Mathematics (Masters) - Sheffield University

Subjects offered:Further Mathematics , Maths+ 3 more

Further Mathematics
Maths
.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!”

£30 /hr

Alex S.

Degree: Physics (Bachelors) - Oxford, St Peter's College University

Subjects offered:Further Mathematics , Physics+ 2 more

Further Mathematics
Physics
Maths
.PAT.

“Studying Physics at the University of Oxford. Looking to tutor maths and physics at all levels, be sure to send me a message if you have any questions!”

£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
Maths
.MAT.
-Personal Statements-
-Oxbridge Preparation-

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

About the author

£24 /hr

Philip D.

Degree: Mathematics (Bachelors) - Exeter University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“Hey, I'm Phil, a mathematics student at the University of Exeter. Unsurprisingly maths has always been a subject that has fascinated me given its intrinsic and logical nature, and I hope that I can help develop your understanding of t...”

You may also like...

Other A Level Further Mathematics questions

Find the modulus-argument form of the complex number z=(5√ 3 - 5i)

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

Find the inverse of a 3x3 matrix

What is the actual use for complex numbers, if they are not really proper number?

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

mtw:mercury1:status:ok