MYTUTOR SUBJECT ANSWERS

894 views

How do you prove that (3^n)-1 is always a multiple of 2?

Proof by Induction: The method of Proof by Induction is a simple but very powerful technique. It involves 3 steps:

1) Showing a claim is true for a basic integer value of n (e.g. 0 or 1)

2) Assuming the claim is true for n=k where k is an arbitrary integer

3) Using this assumption to show that the claim is true for n=k+1

The reason why these 3 steps prove the claim is because you've shown 2 things: You've shown that the claim's true for 0 (or 1). You've also shown that is the claim is true for n=k, then it's also true for n=k+1. And so if the claim is true for n=0 (or 1), then the claim is true for n=1(or 2). Then if the claim is true for n=1 (or 2) then it's true for n=2 (or 3) and so on. So the claim is true for all integers n greater than or equal to 0.

 

1) In this particular case, we'll start with n=0:

(3^0)-1 = 1-1 = 0 = 2 x 0 and so the claim holds.

2) Now let's assume the claim is true for n=k. That is, (3^k)-1 is a multiple of 2. So (3^k)-1 = 2c for some integer c.

3) Now let's look at n=k+1:

(3^(k+1))-1 = 3 x (3^k) - 1

Using our assumption, (3^k)-1 = 2c, so that (3^k)=2c+1

Now we have 3 x (2c + 1) - 1 = 6c + 3 -1 = 6c +2 = 2(3c + 1)

(3^(k+1))-1 = 2(3c + 1) and since (3c + 1) is an integer, we have proven the claim.

Saleem A. A Level Physics tutor, A Level Maths tutor, GCSE Maths tuto...

3 years ago

Answered by Saleem, an A Level Maths tutor with MyTutor


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

424 SUBJECT SPECIALISTS

£20 /hr

Tinashe M.

Degree: Chemical Engineering MEng (Integrated Masters) - Newcastle University

Subjects offered:Maths, Chemistry

Maths
Chemistry

“I love maths, chemistry and biology and my passion for these subjects is something that will help teach students.”

MyTutor guarantee

£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!”

£30 /hr

Mrinank S.

Degree: Engineering (Masters) - Cambridge University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
Further Mathematics
.PAT.
-Oxbridge Preparation-

“Hey there! I'm a second year undergraduate student at the University of Cambridge, studying Engineering. I am really passionate about using science and maths to make the world around us a better place.  I am a patient and friendly tu...”

About the author

Saleem A.

Currently unavailable: for new students

Degree: Mathematics (Masters) - Oxford, St Catherine's College University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
-Personal Statements-
-Oxbridge Preparation-

“I place emphasis on understanding material rather than learning it, you don't need a tutor to help you memorize content”

MyTutor guarantee

You may also like...

Other A Level Maths questions

The point P lies on the curve C: y=f(x) where f(x)=x^3-2x^2+6x-12 and has x coordinate 1. Find the equation of the line normal to C which passes through P.

What is the difference between a definite integral and an indefinite integral?

What is a stationary point on a curve? How do I calculate the co-ordinates of a stationary point?

Do y=3x^2+5x+12 and y=3x-8 intercept with each other? If yes, at which point(s)?

View A Level Maths 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