Binomial Expansion, Sequences and Series Flashcards
Binomial Coefficient
meaning
terms
exponent of a
exponent of b
The coefficient of a term in a binomial expansion of the form (a + b)ⁿ
There are always n+1 terms for a binomial to the nᵗʰ power
Exponent of a decreases by 1 from left to right
Exponent of b increases by 1 from left to right
Binomial theorem
formula
term bʳ
term aʳ
(nC0)aⁿb⁰ + (nC1)aⁿ⁻¹b¹ + (nC2)aⁿ⁻²b² + … + (nCn)a⁰bⁿ
(nCr)aⁿ⁻ʳbʳ
[nC(n-r)]aʳbⁿ⁻ʳ
Finding the coefficient of x given two brackets
Expand the exponential term using binomial theorem,
combine the terms in the brackets with components of the binomial expansion to get the required coefficient then simplify the term to get the coefficient
Finding the value of the constant term independent of x
r is unknown and solve by using the binomial theorem of term b, use the variable values and equate to x⁰ hence to zero, use the r value and multiply by the nCr term to get the value of the constant
Finding the nᵗʰ term in the expansion
n is unknown, expand normally binomially for b and leave answer in (nCr)(x)ⁿ⁻ᶜ
Finding equality for an estimation question
Expand the term with the power corresponding to its approximate expansion, after expanding cancel out each of the power to then solve the equation algebraically to find x. Sub the x value into the given equation that has been binomially expanded.
Types of sequences
A sequence is mapping from the natural numbers to real numbers
Finite sequence - fixed number of terms
Infinite sequence - An ongoing term
Use of natural numbers is important as we have first, second and third terms only and no terms in between
Sequence notation
Each number in a sequence is called a term,
the first term can be given by T₁, t₁, t(1) or T(1)
the general term or nᵗʰ is written as Tₙ or tₙ
Expression of sequences
Listing, listing the terms in order
** use rigid brackets for sequences
Formula for nᵗʰ term
Recursive function, where each term is defined or calculated based on the previous term
Graphing, sequence is illustrated on a graph with the number of the term (n) on the horizontal axis and the term itself (tₙ) on the vertical axis
** Do not joint the points together
Sigma notation
n = 4
Σ 2n + 3
n = 1
Series
Obtained by adding the terms of a sequence of numbers together and can be written in sigma notation, infinite or finite
Arithmetic sequence
Is a sequence were each term is calculated by adding a fixed amount to the previous term,
The first term of an arithmetic sequence is a
The fixed amount or common difference is d
The common difference can be calculated by subtracting a term from the one following it
general term of arithmetic sequences;
tₙ = a + (n-1)d
Arithmetic series
If terms of an arithmetic sequence are added together it forms an arithmetic series
2, 4, 6, 8 - finite
2, 4, 6, 8, … - infinite
2 + 4 + 6 + 8 - arithmetic sequence
2 + 4 + 6 + 8 + … - arithmetic series
Sum of the first n terms of an arithmetic series
Sₙ = 0.5n (2a + [n-1]d)
Equivalent formula using the last term, l
Sₙ = 0.5n[a + l]
tₙ = Sₙ - Sₙ₋₁
Geometric sequence
A sequence where each term is calculated by multiplying the previous term by a fixed number
t₁ = a t₂ = ar t₃ = ar² t₄ = ar³
a is the first term of a sequence
common ratio is r, can be positive, negative an integer or fraction and is calculated by dividing any term by the one before it
r = tₙ₊₁ / tₙ
General term:
tₙ = arⁿ⁻¹
Geometric series
Terms of a geometric sequence added together make a geometric series
(2, 4, 8, 16)* is a finite geometric sequence
* rigid brackets
2 + 4 + 8 + 16 is a finite geometric series
(2, 4, 8, 16, …)* is a infinite geometric sequence
* rigid brackets
2 + 4 + 8 + 16 + … is a infinite geometric series
Sₙ = [ a(rⁿ - 1) ] / [ r -1 ] where r > 1
and
Sₙ = [ a(1 - rⁿ) ] / [ 1 - r ] where r > 1