Basic Sigma Notation and Rules So So r starts at 1 and increases up to a finishing value of n Addition rule: where vr denotes rth term of sequence Constant Multiple rule:

Using the addition and multiplication rules

Changing the signs in a series The term (-1)r means that all the odd terms in the series are –ve The term (-1)r+1 means that all the odd terms in the series are +ve

i.e 5 + 5 + 5 = 15 i.e 1 + 2 + 3 = 6 i.e 12 +22 + 32 = 14 i.e 13 + 23 33 = 36

Difference Method 1) Express the expression g(r) you are trying to sum as g(r) = f(r+1) – f(r) or f(r) – f(r+1) 2) Substitute values from 1 to n into this expression and determine the sum after cancelling the relevant terms. This is in the form g(r) = f(r) - f(r+1) f(r) = f(r+1) =

So g(r) = f(r) – f(r+1) with g(r) = , f(r) = and f(r + 1) =

The terms which remain are 1 - which simplifies to

Questions involving Factorials Exam qu. Show that (r + 2)! – (r + 1)! = (r + 1)2  r! Hence find 221! + 32  2! + 42  3! …………(n + 1)2  n! (r + 2)! = (r + 2)(r + 1)(r )(r – 1)…….. = (r + 2)(r + 1)r! (r + 1)! = (r + 1) (r )(r – 1)…….. = (r + 1)r! So (r + 2)! – (r + 1)! = (r + 2)(r + 1)r! – (r + 1)r! = (r)!(r+1)((r + 2) –1) = r!(r+1)2

= (n + 2)! – 2!