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 nth number in a series: an = a1 + (n – 1) d

Example Given 2, 5, 8, …; find the 100th term

n = 100; a1 = 2; d = 3 an = 2 + (100 – 1) 3 an = 2 + (99) 3 an = 2 + 297 an = 299

Arithmetic Sequences

Summing or adding up n terms in a sequence: Example:

Given 2, 5, 8, …; add the first 50 terms n = 50; a1 = 2; an = 2 + (50 – 1) 3 = 149 Sn = (50/2) (2 + 149) Sn = 25 (151) Sn = 3775

nn aan

S 12

Arithmetic Sequences

Summing or adding up n terms in a sequence: Example:

Given 2, 5, 8, …; add the first 51 terms n = 51; a1 = 2; a2 = 5; an = 2+(51 –

1)3=152 Sn = 2+((51-1)/2) (5 + 152) Sn = 2+25 (157) Sn = 2+3925 Sn = 3927

n 1 2 n

n 1S a a a

2

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 nth number in a series: an = a1 r(n-1)

Example Given 5, 20, 80, 320, …; find the 10th term

n = 10; a1 = 5; r = 20/5 = 4 an = 5 (4(10-1)) an = 5 (49) an = 5 (262144) an = 1310720

Geometric Sequences

Summing or adding up n terms in a sequence: Example:

Given 5, 20, 80, 320, …; add the first 7 terms n = 7; a1 = 5; r= 20/5 = 4 Sn = 5(1 – 47)/(1 – 4) Sn = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3) Sn = (– 81915)/(– 3) = (81915)/(3) Sn = 27305

r

raS

n

n

1

11

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