Sequences and series - Topic 4 Flashcards
Binomial expansion, nth term, sigma notation, arithmetic sequences, geometric sequences, and modelling
Year 1 - Chapter 8.2
Equation for factorial notion?
ⁿCᵣ = n!/r!(n-r)!
Year 1 - Chapter 8.3
What is binomial expansion?
(a+b)ⁿ =
aⁿ + (n|1)aⁿ⁻¹b + (n|2)aⁿ⁻²b² +… + (n|r)aⁿ⁻ʳbʳ + bⁿ
Year 2 - Chapter 4.1
How to do binomial expansion for (1+x)ⁿ
(1+x)ⁿ = 1 + nx + (n(n-1)/2!)x² + ((n(n-1)(n-2))/3!)x³ +… + (n(n-1)…(n-r+1)/r!)xʳ +…
Only valid for |x|< 1, nER
Year 2 - Chapter 4.2
How to do binomial expansion for (a+bx)ⁿ
(a+bx)ⁿ = (a(1+(b/a)x))ⁿ = aⁿ(1+(b/a)x)ⁿ
Then follow the same structure as (1+x)ⁿ
n must be a negative or a fraction
is valid for |(b/a)x| < 1 or |x| < |a/b|
Year 2 - Chapter 3.1
Formula for an arithemtic sequence?
uₙ = a + (n-1)d
a is the first term
d is the common difference
uₙ is the nth term
Year 2 - Chapter 3.2
Equation for the sum of the first n terms in an arithmetic sequence?
Sₙ = n/2 x (2a + (n-1)d)
Year 2 - Chapter 3.3
Equation for a geometric sequence?
uₙ = arⁿ⁻¹
a is the first time
r is the common ratio
Year 2 - Chapter 3.4
Equation for the sum of the first n terms in an geometric sequence?
Sₙ = (a(1-rⁿ))/(1-r); r<1
Sₙ = (a(rⁿ-1))/(r-1); r>1
r ≠ 1
Year 2 - Chapter 3.5
Equation for sum to infinity of a convergent geometric sequence?
S∞ = a/(1-r)
Only for |r|<1
Year 2 - Chapter 3.6
What does this symbol mean: Σ?
The sum of all terms. The limits of the sequence are written on top of Σ, and the r term to know from where you start summing from.
Year 2 - Chapter 3.7
How do recurrence relations work?
A recurrence relation of the form uₙ₊₁ = f(uₙ) defines each term of a sequence as a function of the previous term.
Example:
- uₙ₊₁ = 2uₙ + 3, u₁ = 6 → u₂ = 2(6) + 3 = 15
Year 2 - Chapter 3.7
How do you know whether a sequence is increasing or decreasing?
uₙ₊₁ > uₙ; increasing sequence
uₙ₊₁ < uₙ; decreasing sequence
uₙ₊ₖ = uₙ; periodic sequence
