• Proof by induction

To prove by induction that a statement (C) involving an integer n is true for all [pic], prove that, i) (C) is true for [pic];

ii) if (C) is true for [pic]then (C) is true for [pic].

And then bring these parts together to complete the proof.

• Summing series using formulae

To sum series, use a combination of the following formulae. Second two are given in formula book. [pic][pic] [pic]

e.g. Q: Find [pic]

By substituting in formulae find,

[pic]

• Summing series using method of differences

e.g. Q: Find [pic]

A: First split into partial fractions [pic]

Write out the first three terms and last two or three terms. Last term will be the one where you substitute in r=n. [pic]

See what cancels

[pic]

• Properties of the roots of polynomial equations

If [pic] and [pic] are the roots of quadratic equation [pic], then [pic]

[pic]

If [pic], [pic] and [pic] are the roots of cubic equation [pic], then [pic]

[pic]

[pic]

If [pic], [pic], [pic] and [pic] are the roots of quartic equation [pic], then [pic]

[pic]

[pic]

[pic]

• Finding equations with related roots

Q: The roots of the cubic equation [pic] are [pic], [pic] and [pic] where [pic]. Find the cubic equation with roots, [pic]. A: Method 1

(Let ‘ denote new coefficient)

[pic]

[pic]

[pic]

Let [pic] [pic] and [pic]

So [pic]

Method 2 (By substitution)

Let [pic], then [pic] are roots of [pic] iff [pic] are the roots of [pic] So [pic]

• Sketching graphs of rational functions

To sketch the graph of [pic]:-

1. Find the intercepts i.e. substitute in [pic] (to find y intercepts) and make [pic] (to find x intercepts). 2. Find the asymptotes (a) Vertical, when [pic]

(b) Horizontal,

To find horizontal asymptotes complete a long division to write the fraction as a sum of a polynomial and a proper fraction. Then examine the behaviour as x tends to infinity. (This only needs to be completed when the fraction was improper) 3. Consider nature of graph as it tends to vertical asymptote. 4. Consider the nature as [pic].

5. Complete the sketch (Check on graphical calculator)

n.b. Can also use information about turning points.

• Basic complex number arithmetic

Use usual algebra manipulation combined with [pic]

e.g. Q: [pic] , [pic] and [pic]

Find (i) [pic](ii) [pic]

(iii) [pic](iv) [pic]

A: (i) [pic]

(ii) [pic]

(iii) [pic]

(iv) [pic]

[pic]

Compare real parts [pic] Compare imaginary parts [pic]

Solve simultaneously [pic] [pic] [pic]

Rearrange to get [pic] Quadratic in terms of [pic], solve using quadratic formula to get [pic] or –4 Require b to be real [pic][pic]

Substitute in for [pic]

[pic] or [pic]

• Solving linear equations with complex coefficients

Use ordinary algebraic manipulation, combined with the fact that two complex numbers are only equal if both the imaginary and real parts are equal. e.g. Q: Solve the following equations, where [pic] and w are complex numbers. [pic][pic] A: [pic](x2)[pic](x3)

[pic][pic]

First equation take the second equation gives,

[pic]Let [pic]

[pic]

Equate real and imaginary parts [pic] and [pic]

Solve simultaneously to find [pic] and [pic]

[pic]

To find z substitute value of w into original equation and let [pic] [pic] Equate real and imaginary parts to find,

[pic]

• Solving quadratics with real coefficients

Use quadratic formula and don’t forget to substitute in [pic]

• Solving higher order polynomials with real coefficients Use the following result:-

If z is a complex root of [pic], where f is a...