Hello everyone - the name is Shay. I am a third year mathematics student at the University of Warwick. I sat my A-Levels in 2015 (Maths, Further Maths, Economics, Chemistry and French) and should be able to help with Maths, Further Maths & Economics questions (all at GCSE and A-Level); GCSE French questions; maths university admissions tests (MAT, STEP & AEA) and UCAS Personal Statements. My objective is simple - to ensure my tutee understands the ideas behind the key ideas of topics. I will do my best to help people understand what's going on in a concise and enjoyable manner. I have some mentoring experience at school as well as tutoring on and outside the MyTutor platform.

If you have any questions, send me a 'WebMail' or book a 'Meet the Tutor Session'! (both accessible through this website). Remember to tell me your exam board and what you're struggling with - I can't help you if I don't know what to help you with!

I look forward to meeting you!

Hello everyone - the name is Shay. I am a third year mathematics student at the University of Warwick. I sat my A-Levels in 2015 (Maths, Further Maths, Economics, Chemistry and French) and should be able to help with Maths, Further Maths & Economics questions (all at GCSE and A-Level); GCSE French questions; maths university admissions tests (MAT, STEP & AEA) and UCAS Personal Statements. My objective is simple - to ensure my tutee understands the ideas behind the key ideas of topics. I will do my best to help people understand what's going on in a concise and enjoyable manner. I have some mentoring experience at school as well as tutoring on and outside the MyTutor platform.

If you have any questions, send me a 'WebMail' or book a 'Meet the Tutor Session'! (both accessible through this website). Remember to tell me your exam board and what you're struggling with - I can't help you if I don't know what to help you with!

I look forward to meeting you!

In my sessions, I will try to make maths and/or economics seem enjoyable to you - while many of you may not believe this, but these subjects are really amazing. A lot can be done in a fun hour - there will be some jokes and banter, but I will strive to ensure my tutee understands the material. I will use as many different ways (diagrams, words, analogies, poor jokes - my puns, despite what my friends say, are in my opinion unrivalled) as possible to explain a concept, until you are confident enough to explain what on earth is going on to me - if not me, you can always try the dog!

In my sessions, I will try to make maths and/or economics seem enjoyable to you - while many of you may not believe this, but these subjects are really amazing. A lot can be done in a fun hour - there will be some jokes and banter, but I will strive to ensure my tutee understands the material. I will use as many different ways (diagrams, words, analogies, poor jokes - my puns, despite what my friends say, are in my opinion unrivalled) as possible to explain a concept, until you are confident enough to explain what on earth is going on to me - if not me, you can always try the dog!

No DBS Check

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*

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*

Competitiveness is what is says on the tin really - if there are many firms in an industry each with a similar degree of market power, the market is competitive.

Contestability is something a little more subtle. It is the perceived threat of competition. For example, an industry may be an oligipoly, where there are only a few firms each with a large degree of market power. However, if there are low barriers to entry (i.e. low sunk costs, low costs of red tape) - basically if it's easy for new firms to enter the industry - incumbent firms will act as if there is actual competition. Contestability is all about the perceived threat of new firms entering the market and taking away some of the market power of incumbent firms.

Competitiveness is what is says on the tin really - if there are many firms in an industry each with a similar degree of market power, the market is competitive.

Contestability is something a little more subtle. It is the perceived threat of competition. For example, an industry may be an oligipoly, where there are only a few firms each with a large degree of market power. However, if there are low barriers to entry (i.e. low sunk costs, low costs of red tape) - basically if it's easy for new firms to enter the industry - incumbent firms will act as if there is actual competition. Contestability is all about the perceived threat of new firms entering the market and taking away some of the market power of incumbent firms.