Sequences + Series

Arithmetic sequence

AKA linear sequence or arithmetic progression, is one in which the change is always constant

What formula is used to prove a sequence is arithmetic?

Tn - (Tn - 1) = constant

What is an arithmetic sequence in terms of a and d?

a, a +d, a +2d, a +3d, …

3x + 1, 5x + 1, 8x + 1, …
is an arithmetic sequence
find x

let d = d
simplify
solve for x

Quadratic sequence

one in which the second difference is a non-zero constant

How do you find the difference in a quadratic sequence?

let 2a = second difference
solve for a

How do you find the general term for a quadratic sequence?

Tn = an² + bn + c
find a
let n = 1 and form an equation
let n = 2 and form another equation
do simultaneous equations to find b and c

How do you find the term number when given a term in a quadratic sequence?

let the term equal the general term
form a quadratic equation = 0
solve for n
n cannot be negative

Cubic sequence

one in which the third difference is a non-zero constant

Geometric sequence

one in which each term is found my multiplying the previous term by a constant called the common ratio

What formula is used to prove a sequence is geometric?

Tn/Tn - 1 = constant

What is a geometric sequence in terms of a and r?

a, ar, ar², ar³, ar⁴, …

How do you find the common ratio/r?

T₂/T₁

What does ∑ mean?

add from 0 - ∞

Recursive sequence

one in which you need to know the previous term to get the next term in the sequence

What symbol expresses that a series should be added from T₁ to T₁₀

When is a(rⁿ - 1)/r - 1 used

when r > 1

2x - 4, x + 1, x - 3
is a geometric sequence
find the two possible x values

let r = r
form a quadratic equation
solve for x
x = 1 x = 11

T₃ = 32 T₅ = 512
find r and a

find the terms in terms of a and r
divide the lines into eachother, the a's will cancel
solve for r = ±4
sub in r, solve for a = 2

How do you find a limit when x → R?

sub in value for x
finish equation
if equal to 0/0, the equation has a difference of two squares or is cubed
factorise the equation, simplify
sub in value for x
find limit by finishing equation

How do you find the limit when n → ∞

divide every term by the highest power of n
terms that still have n as denominator are so small they can be considered 0
solve

What is a convergent limit?

when the limit is o or o/number

What is a divergent limit?

when the limit is number/0
not a real number