Laws of Indices
- a^m x a^n =a^m+n If you multiply two numbers, you added the powers.
- a^m/a^n =a^m-n If you divide two numbers, you subtract thier powers.
- (a^m)^n= a^mn If you have a power to the power of something else, multiply the powers together.
- a^1/m=ma You can write roots as powers
- a^m/n = na^m A power thats a fraction is the root of the power e.g. 16^3/4 = (16^1/4)^3 = 8.
- a^-m= 1/a^m A negative power means the denominatior of a fraction.
- a^0= 1
starting with y=f(x)
- y=f(x)+a Adding a number to the whole function translates the graph in the y-direction. 1) If a>0, the graph goes upwards. 2) If a<0, the graph goes downwards.
- y=f(x+a) Writing 'x+a' instead of 'x' means the graph moves sideways (translated in the x-direction) 1) If a>0, the graph goes to the left. 2) If a<0, the graph goes to the right.
Streches and Reflections
- y=af(x) Negative values of 'a' reflects the basic shape in the x-axis. If a>1 or a<-1 the graph is streched vertically. If -1<a<1 the graph is squashed vertically.
- y=f(ax) Negative values of 'a' reflect the basic shape in the y-axis. If a>1 or a <-1 the graph is squashed horizontally. If -1<a<1 the graph is stretched hoizontally.
A sequence can be defined by its nth term. For the sequence 7,12,17,22,27, Un= 5n+2 because difference of 5 between each term then to get from 5 to 7 you have to add 2.
- ak just means the Kth term of the sequence.
- Example find the recurrence relation of the sequences 5,8,11,14,17... The common difference is 3, ak+1=ak+3, so if k=5 then ak=a5 with ak+1=a6 but the description needs to be more specific, you have got to give the first term in the sequence as well as the recurrents relation. Answer: ak+1=ak+3 when a1=5
- Example 2, A sequence is defined by ak+1=2ak-1, a2=5. List the first 5 terms. a3=2x5-1=9, a4=17, a5=33, a1= a2=2a1-1, 5=2a1-1, 6=2a1, 3=a1 therefore the first 5 terms are 3,5,9,17,33.
Types of Sequences
- E.g.ak+1=ak+3 a1=1, 1<k<20 will contain 20 terms meaning it is an finite sequence.
- An infinite sequence wont have a final turm
- Others revisit the same numbers over and over in a period, called a periodic sequence.
- In some sequences the terms get closer to a limit however they never reach it. ak+1=f(ak). the limit can be found by solving L=f(L).
- If a sequence doesnt approch a limit is a divergent sequence.
- Example for convergent sequence: A sequence is defined by the relation ak+1=-1/2ak+2. Find L, the limit of ak as k tends to infinity. L=-1/2L+2, 3/2L=2, L=4/3.
- 1st term: a
- Common difference: d
- Position of term: n.
- nth term: a+(n-1)d
- Sum of progession: nx(a+l)/2
- If you don't know l sn= n/2[2a+(n-1)d]
Sigma also means sum of the series. the number below is your starting nth term and the last term is above
- The sum of the first n natural numbers (positive whole number) looks like this: Sn=1+2+3+...+(n-2)+(n-1)+n so a=1, l=n and n=n. Sn= 1/2n(n+1)
Geometric Progressions and Series
- Sn=a[(1-r^n)/1-r] which is the same as a[(r^n -1)/r-1]
- The sum to infinaity=
- A divergent series doesn't have a sum to infinity as its total can be worked out.
Sloving Aritimetic and Geomentric Progressions.
- Simultaneous Equations,
- Quuadratic Equations,
- Using the Equations.
- Pasels Triangle shows patterns in the coefficients by adding the 2 terms above to get the next row.
- to find the x coffient to a power use the formaula nCrx (a)^r x (b)^n-r for brackets (a+b).
- If it ask for the coefficient of x^4 you don't need to but the x in the answer.
- if have two brackets find all the ways in making the asked for coefficient then added the possible ways together to find the total coefficient.
Trig Formulas to Know.