Topic 1 (HL): Algebra
1.1 Arithmetic sequences and series
Sequence
Set of quantities arranged in a definite order, for instance 1, 2, 3, 4... or 1, 2, 4, 8...

un  nth term, u1  first term, u2  second term...
Series
Result of the addition of the terms of a sequence.

Sigma notation (∑): The sum of...
Arithmetic

Common difference (d): Difference (d) between consecutive terms is always the same.

d = un  un1


Sequence: un = a + (n  1)d; a = u1 = first term

Series: Sum of the n terms = Sn = n/2 [2a + (n1)d] = n/2 (u1 + un)
Geometric

Common ratio (r): Common ratio (r) between consecutive terms is always the same.

r = un/un1


Sequence: un = a x r^n1

Series: Sum of the n terms = Sn = a (1  r^n)/(1  r), │r│< 1 or Sn = a (r^n  1)/(r 1), │r│> 1

Convergent series: Geometric series with │r│< 1 (terms get smaller as n increases)

Sum of all sequence's terms will tend to approach a certain value: S∞ = a/(r 1)


Applications

Compound interest: $ invested at r % p. a. (per annum) compound interest

An = P (1 + r/100)^n, where An is the amount after n time periods


Superannuation: Investing each year the same amount in a scheme

Total money at the end of the scheme is the sum of each investment done each year

Sn = a (r^n  1)/(r 1) = P (1 + r/100)[1  (r/100)^n]/[1  (1 + r/100)]; P = initial investment

1.2 Exponents and logarithms
Exponents rules
Base e
Euler's number (e): An irrational number (like ) found in natural patterns.
Exponential function
f(x) = a^x, x ϵ R, a > 0 and a ≠ 1

Graphs with a > 1 (yvalues increase as xvalues increase)

Called "exponential growth"

Cases: When x < 0 then 0 < y < 1; When x>0 then y > 1; At x = 0, y = 1

Horizontal asymptote at y = 0


Graphs with 0 < a < 1 (yvalues decrease as xvalues increase)

Called "exponential decay"

f(x) = a^x = (1/a)^x


Horizontal translation: y = a^(x + k); Vertical translation: y = a^x + k; Dilation along yaxis: y = ka^x
Logarithms
y = b^N means that N = logb(y), y ≥ 0 and b ≠ 1 and loge(x) = ln(x)
Logarithm rules
Logarithmic function
Reflection of the exponential upon the line y = x

Horizontal translation: y = log a (x + k)

+k: Assymptote at k

k: Assymptote at +k


Vertical translation: y = log a (x) + k

Stretching and shrinking along the yaxis: y = k log a (x)

If k<0: Reflection about xaxis


Reflection about yaxis: y = log a (x)
Examples (not IB!)

Richter scale: Magnitude of an earthquake

R(magnitude) = log 10 (I/Io)


Ehrenberg relation: Measurement of a children's weight.
1.3 Counting principles and binomial theorem
Counting principles
Permutation (Arrangement): Counting process where the order must be taken into account (AC ≠ CA).

Rule 1: If any one of n different mutually exclusive and exhaustive events can occur on each of k trials, the number of possible outcomes is equal to n^k.

Example: Dice rolled twice 6^2 = 36 events.


Rule 2: If there are n1 events on the first trial, n2 events on the second trial and so on, and finally, nk on the kth trial, then the number of possible outcomes is equal to n1 x n2 x ... x nk.

Example: Three different pairs of pants, four different shirts and five different shirts 3x4x5 = 60 ways to dress


Rule 3: The total number of ways that n different objects can be arranged in order is equal to n! (factorial) = n x (n1) x (n2) x ... x 3 x 2 x 1 (0! = 1)

Example: In how many ways can seven children sit on a park bench? 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 arrangements.


Rule 4: The total number of ways of arranging n objects, taking r at a time is given by nPr = n!/(nr)!

Example: Total number of arrangements of 8 books on a bookshelf if only 5 are used is given by 8P5 = 8!/(85)!

Box Method: Filling available spaces, represented by boxes.


Rule 5: The number of permutations of n objects of which n1 are identical, n2 are identical, ..., nk are identical is given by n!/(n1! x n2! x ... x nk!)

Useful for anagrams with words.

Combination (Selection): Counting process where the order has no importance (AC = CA).

Rule 6: The total number of ways of selecting n objects, taking r at a time, for instance, a selection of 5 books from 8 different books, is given by:
Binomial theorem
Binomials: Expressions made out of 2 terms (bi), e.g. (a + b)^2

Powers of a: Start at n and decrease a^n, a^(n1), a^(n2), a^(n3), ..., a^0

Powers of b: Start at 0 and increase: b^0, b^1, b^2, b^3, ..., b^n

Coefficients follow Pascal's triangle: Numbers in the body are found by adding the two numbers immediately above to either side

Example: (x +4)^4 = 1x^4 + 4(x^3)a +6(x^2)(a^2) + 4(x)(a^3) + 1a^6

1 4 6 4 1 corresponds to the 4th row in the Pascal's triangle.


If there is a minus sign (), it alternates between the coefficients: + ...  ... + ...  ...
The general term: (x + a)^n = (