MATH 2160 Sequences

Arithmetic Sequences The difference between any two consecutive terms is always the same. Examples: 1, 2, 3, … 1, 3, 5, 7, … 5, 10, 15, 20, … Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …

Arithmetic Sequences The n th number in a series: a n = a 1 + (n – 1) d Example Given 2, 5, 8, …; find the 100 th term n = 100; a 1 = 2; d = 3 a n = 2 + (100 – 1) 3 a n = 2 + (99) 3 a n = a n = 299

Arithmetic Sequences Summing or adding up n terms in a sequence: Example: Given 2, 5, 8, …; add the first 50 terms n = 50; a 1 = 2; a n = 2 + (50 – 1) 3 = 149 S n = (50/2) ( ) S n = 25 (151) S n = 3775

Arithmetic Sequences Summing or adding up n terms in a sequence: Example: Given 2, 5, 8, …; add the first 51 terms n = 51; a 1 = 2; a 2 = 5; a n = 2+(51 – 1)3=152 S n = 2+((51-1)/2) ( ) S n = 2+25 (157) S n = S n = 3927

Geometric Sequences The ratio between any two consecutive terms is always the same. Examples: 1, 2, 4, 8, … 1, 3, 9, 27, … 5, 20, 80, 320, … Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …

Geometric Sequences The n th number in a series: a n = a 1 r (n-1) Example Given 5, 20, 80, 320, …; find the 10 th term n = 10; a 1 = 5; r = 20/5 = 4 a n = 5 (4 (10-1) ) a n = 5 (4 9 ) a n = 5 (262144) a n =

Geometric Sequences Summing or adding up n terms in a sequence: Example: Given 5, 20, 80, 320, …; add the first 7 terms n = 7; a 1 = 5; r= 20/5 = 4 S n = 5(1 – 4 7 )/(1 – 4) S n = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3) S n = (– 81915)/(– 3) = (81915)/(3) S n = 27305

The Ultimate Pattern… Fibonacci Sequence

Rabbit Breeding Pattern (# of Pairs)

The Golden Rectangle

The Golden Ratio

Fibonacci Sequences 1, 1, 2, 3, … Seen in nature Pine cone Sunflower Snails Nautilus Golden ratio (n + 1) term / n term of Fibonacci Golden ratio ≈ 1.618