# Core 2

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## 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
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## Graph Transformation

starting with y=f(x)

Translations

• 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.
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## Sequences 1

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.

Recurrance relation

• 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.
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## Sequences 2

Convergent Sequences

• 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.
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## Arithmetic Progessions

• 1st term: a
• Common difference:
•  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)
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## Geometric Progressions and Series

• Un=ar^n-1
• 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,
• Using the Equations.
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## Binomial Expansions

• 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.
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Area=0.5absinC

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