MYTUTOR SUBJECT ANSWERS

117 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...

3 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

62 SUBJECT SPECIALISTS

£20 /hr

Isabel R.

Degree: Mathematics (Bachelors) - Manchester University

Subjects offered: Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“I am currently a first year studying Mathematics at the University of Manchester-so A Levels and GCSEs are still fresh in my mind when it comes to remembering how I learnt the material myself. In school I mentored GCSE students in Mat...”

MyTutor guarantee

£22 /hr

Lloyd S.

Degree: Mathematics G100 (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics
Maths

“About Me:I am 19 and from Devon, currently in my first year studying Maths at the University of Bristol. I have a real passion for Maths and I really hope I can help you to understand, and maybe even enjoy doing maths - I know not ev...”

£20 /hr

Ross G.

Degree: Mathematics (Masters) - Durham University

Subjects offered: Further Mathematics , Physics+ 2 more

Further Mathematics
Physics
Maths
.STEP.

“I am a maths student at Durham University. I love maths and I love studying it, both now and at A levels. I know that maths isn’t for everybody but I hope that in my tutorials I can help you learn the content and, maybe, you can learn ...”

About the author

£20 /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

What is sin(x)/x for x =0?

How do I sketch the locus of |z - 5-3i | = 3 on an Argand Diagram?

How to use the integrating factor?

How to determine the rank of a matrix?

View A Level Further Mathematics tutors

Cookies:

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

mtw:mercury1:status:ok