Proof+by+Induction

__**Proof by Induction**__ 7th September

Basic idea of Induction - (i) show that the statement holds when n = 1, (ii) Assume the statement true for n = k. Write down this statement with k in it! (iii) Consider the case where n = k + 1. Use the assumption (the statement you wrote down) to simplify this case. Show that it is simply the statement you wrote down earlier with k replaced by (k + 1) (iv) Don't forget those key words at the end... BUT, this is the earlier statement with k replaced by (k + 1) Therefore, if the original statement is true for n = k it is true for n = k + 1 Since it is true for n = 1 it is true for all positive integer values of n.

Here is today's lesson:

We looked at a couple of exam questions: June 2005 q6 and Jan 2006 q6

10th September More proof by induction...we've got to be very careful with algebraic manipulation. FACTORISE whenever possible!

Although a proof by induction question will always follow the same 'pattern' as set out above, sometimes the actual reasoning may be a bit different. Many examples will involve the sum of a series (in which case we just add another term to the series and simplify) but in other examples we may need, for example, to show that a given formula is always divisible by, say, 9. This can be a bit trickier. Always use the 'answer you're trying to find' to help! Here is today's lesson: Homework (due in next Monday) is included!


 * Wednesday 12th September**

'Method of differences'

This is an alternative method of proving that a series has a general formula, or finding a specific sum. Start by showing that a given expression may be written as the difference of two terms (sometimes a combination of more than two terms). Then show that when you sum these other terms most of them cancel out, leaving a small number (hopefully just two - one from the first expresion and one from the last, but sometimes more)

Lesson:


 * Friday 14th September**

Sums of specific series. Formulas for the sum of the first n square numbers, and the sum of the first n cube numbers are given in the formula book. You need to know that the sum of the first n integers is given by 1/2 n(n + 1), and that 'sigma 1' (for r from 1 to n) is just n.

SORRY - I lost that lesson!