Sequences

A finite sequence: 1, 3, 5, …, 91

An infinite sequence: 1, 3, 5, …

n = position in the sequence (index) STARTS AT 1

• $Tn$ = the nth term

• difference between arithmetic, geometric sequences etc. is the progressions between the terms i.e. the definition of each term

Arithmetic Sequences:

("Arithmetic Progressions" or "APs")

e.g: 1,2,3,4 or 3,6,9,12

For an arithmetic sequence:
$T_n=T_{n-1}+d$
$T_1=a$
d: constant term (the common difference)
a: constant term (first term)

• if a sequence is arithmetic, $T_n-T_{n-1}$ gives the same result for all pairs of successive terms

• Any arithmetic progression can be written as:

  • a, (a+d), (a+2d), (a+3d), …
  • Where: $$
    T_{n}=a+(n-1)d

Example questions: ![[Pasted image 20231008122932.png|400]] ## Geometric Sequences: `("Geometric Progressions" or "GPs")` e.g.: `2, 4, 8, ......` ^ T1 = 2, r = 2 Recursive Definition: For a geometric sequence: $T_{n}=(T_{n-1})r$ $T_{1}=a$ r: constant (the common ratio) - if a sequence is geometric, $\frac{T_{n}}{T_{n-1}}$ gives the same result for all pairs of successive terms - Any geometric progression can be written as: - a, ar, ar^2, ar^3 - Where: $$ T_{n}=ar^{n-1}