MYTUTOR SUBJECT ANSWERS

433 views

How does proof by mathematical induction work?

Mathematical induction is a way of proving statements in maths. The principle is quite similar to dominoes (not pizza, the game) - if you push the first one, the second one will be pushed over, pushing the third one and so on. 

There are 3 stages to induction. The first stage is to prove that the base case, n = 1 is true (essentially the first domino). The second stage involves you assuming the case n = k is true. The final stage involves you using the assumption above to show that the case n = k + 1 is also true (essentially you're showing that if you push the kth domino, the (k + 1)th domino will also be pushed over).

Here is an example of proof by induction being applied:

For any positive integer n, 1 + 2 + 3 + ... + n = n(n + 1)/2.

Let P(n) be the assumption that 1 + 2 + 3 + ... + n = n(n + 1)/2. Consider the base case, n = 1:

Left-hand side = 1. Right-hand side = 1(1 + 1)/2 = 1
As LHS = RHS, P(1) is true.

Assume that P(k) is true i.e. 1 + 2 + 3 + ... + k = k(k + 1)/2. Consider P(k + 1) [remember we have to use P(k) somewhere here]

1 + 2 + 3 + ... + k + (k + 1) = k(k + 1)/2 + (k + 1) (using P(k))
Taking a factor of (k + 1) from the RHS:
k(k + 1)/2 + (k + 1) = (k + 1)[k/2 + 1] = (k + 1)(k + 2)/2 = (k + 1)((k + 1) + 1)/2 i.e. P(k + 1) is true.

Therefore, as P(1) is true and P(k) true implies P(k + 1) is true, by the principle of mathematical induction, 1 + 2 + 3 + ... + n = n(n + 1)/2

Shayantan C. A Level Maths tutor, A Level Further Mathematics  tutor,...

1 year ago

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


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

86 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...”

£30 /hr

Thomas D.

Degree: Economics (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths+ 2 more

Further Mathematics
Maths
Geography
Economics

“Why choose me? I am a highly academic student having achieved 4 A* at A level but also have excellent communication and mentoring skills havingtutored fellow students at school and beenHead Boy. I am honest, reliable and precise, if ...”

£20 /hr

Katherine H.

Degree: Mathematics (Masters) - Bath University

Subjects offered: Further Mathematics , Maths+ 1 more

Further Mathematics
Maths
Geography

“I am a first-year undergraduate for a masters degree in Mathematics at the University of Bath. I achieved full UMS marks in Mathematics (600/600), Further Mathematics (600/600), Geography and Psychology A Levels. I have tutored on num...”

About the author

Shayantan C.

Currently unavailable: for regular students

Degree: MMath (Bachelors) - Warwick University

Subjects offered: Further Mathematics , Maths+ 5 more

Further Mathematics
Maths
French
Economics
.STEP.
.MAT.
-Personal Statements-

“Hello everyone - the name is Shay, and I'm doing a maths degree at the University of Warwick. I did my A-Levels last year (Maths, Further Maths, Economics, Chemistry and French) and should be able to help with A-Level Maths, Further M...”

You may also like...

Posts by Shayantan

How does proof by mathematical induction work?

What is the difference between competitiveness and contestability?

Other A Level Further Mathematics questions

Find the general solution of: y'' + 4y' + 13y = sin(x)

How do I use proof by induction?

if y = (e^x)^7 find dy/dx

Find the inverse of the general 2x2 matrix A= ([a, b],[c, d]) when does this inverse exist?

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