Join TSR now and get all your revision questions answeredSign up now

Induction problem

    • Thread Starter
    Offline

    2
    ReputationRep:
    Hi everyone,

    I've been given the statement "9^n -1is a multiple of eight for any positive integer n"

    I know that 9^1 - 1 = 8

    And 9^k - 1 = 8r (where r is a natural positive number,

    But I'm having trouble with 9^k + 1... How would I break this up to show that this too can be written as a multiple of 8?
    Offline

    3
    ReputationRep:
    Don't want to give too much away, so here's an explicit example when k=3. You'll need to generalise the approach.

    9^3 - 1 = 728 = 91 * 8

    Note that 9^3 = 91 * 8 + 1.

    So 9^4 = 9 * (91 * 8 + 1) = 9 * 91 * 8 + 9
    So 9^4 - 1 = 9 * 91 * 8 + 8

    Which is obviously a multiple of 8.
    • Thread Starter
    Offline

    2
    ReputationRep:
    Thanks - Also, can I have some help with the following proof - I don't think you need to use induction:

    If p is a prime number such that p >3, then p^2 - 1 is a multiple of 24

    I'm just not sure where to start really.
    Offline

    3
    ReputationRep:
    (Original post by Electrogeek)
    Thanks - Also, can I have some help with the following proof - I don't think you need to use induction:

    If p is a prime number such that p >3, then p^2 - 1 is a multiple of 24

    I'm just not sure where to start really.
    Two hints:

    Factorise it.

    If you multiply three consecutive integers together, which numbers do you know will definitely be factors of the product?
    Offline

    3
    ReputationRep:
    Factorise p^2 - 1, consider what it means to be a multiple of 24.

    Spoiler:
    Show
    The fact that p is a prime > 3 is actually a lot stronger than you need here. The result is true for a lot of other numbers too.

    Spoiler:
    Show
    So if you get stuck, one thing you might want to do is simply go through n = 1, 2, 3, ..., 50 and for each n note if n^2-1 is a multiple of 24 and see if you can see a pattern.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by tiny hobbit)
    Two hints:

    Factorise it.

    If you multiply three consecutive integers together, which numbers do you know will definitely be factors of the product?
    That really helps - I don't know why I didn't see it myself! Thanks for the hints.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by TeeEm)
    Have you come across the generalised identity an - bn = (a - b)( .......)?
    Yeah we've done difference in two squares. I've managed to prove it now, but thanks for the help anyway.
 
 
 
Poll
Do you have exam superstitions?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.