{1, 1, 2, 3, 5, 8, 13, 21,...} What is this? Fibonacci Sequence

Chapter 13: Sequences and Series -Identify an arithmetic or geometric sequence -Use recursive and explicit formulas A sequence is an ordered set of numbers. Each number in the sequence is a term of the sequence. A sequence may be infinite (without end) or finite (limited number of terms). Arithmetic sequence: the difference between each term is constant. Ex: {2, 6, 10, 14, 18,... } Difference = 4 Ex: {17, 10, 3, -4, -11, -18,... } Difference = -7 Geometric sequence: the ratio between any two consecutive terms is constant. Ex: {1, 3, 9, 27, 81,... } Ratio = 3 Ex: {64, -32, 16, -8, 4,... } Ratio = -1/2 Neither: sometimes a sequence is neither arithmetic nor geometric. Ex: {1, 4, 9, 16,... } n2

Arithmetic or Geometric? {17, 21, 25, 29,... } (8, 12, 18, 27,... } {15, 7, -1, -9,... } {4, 8, 16, 32,... } Arithmetic, +4 Geometric, 3/2 Arithmetic, -8 Geometric, 2

A recursive formula is a rule in which one or more previous terms are used to generate the next term. An explicit formula defines the nth term of a sequence as a function of n which is useful when you do not know a term's preceding term. Arithmetic Example: {7, 5, 3, 1, -1,... } {a1=7, a2=5, a3=3, a4=1, a5=-1,... } recursive formula an = an explicit formula an = 9 - 2n

Geometric Example: {20, 40, 80, 160, 320,... } {a1=20, a2=40, a3=80, a4=160, a5=320,... } recursive formula an = 2an-1 explicit formula an = 10(2n)

Tower of Hanoi Puzzle

Arithmetic Sequence a1, a1 + d, a1 + 2d, a1 + 3d, a1 + 4d,... where, a1 = first term and d = common difference Recursive an=an-1+d Explicit an = a1 + (n-1)d or an = a1 - d +dn Partial Sum (for n terms) Sn = ½n (a1 + an)

Arithmetic Example: Find the recursive and explicit formulas for the sequence {7, 11, 15, 19,... } Recursive an = an-1 + d Explicit an = a1 + (n-1)d or an = a1 - d +dn d = 4, therefore an = an d = 4, therefore an = 7 + (n-1)4 which simplifies to an = 7 + 4n - 4 = 3 + 4n or an = n which also simplifies to 3 + 4n So.. an = 3 + 4n

Arithmetic Example: Find the recursive and explicit formulas for the sequence {9, 4, -1, -6, -11,... } Recursive an = an-1 + d Explicit an = a1 + (n-1)d or an = a1 - d +dn d = -5, therefore an = an d = -5, therefore an = 9 + (n-1)(-5) which simplifies to an = 9 - 5n + 5 = n or an = 9 - (-5) - 5n which also simplifies to n So.. an = n

Arithmetic Example: Find the first six terms and the 300th term of the arithmetic sequence 13, 7,... Recursive an = an-1 + d Explicit an = a1 + (n-1)d or an = a1 - d +dn d = -6, therefore an = an-1 - 6, but doesn't help much! d = -6, therefore an = 13- (-6) - 6n which also simplifies to n So.. an = n a1=13, a2=7, a3=1, a4=-5, a5=-11, a6=-17 a300 = (300) = = -1781

Arithmetic Example: Find the SUM of the first 40 terms of the arithmetic sequence 3, 7, 11, 15,... Recursive an = an-1 + d Explicit an = a1 + (n-1)d or an = a1 - d +dn d = 4, therefore an = an-1 = 4, but doesn't help much! d = 4, therefore an = n which simplifies to n So.. an = n a40=-1 + 4(40) = = 159 Partial Sum (for n terms) Sn = ½n (a1 + an) S40 = ½40 ( ) =20(162) = 3240

Arithmetic Example: Find the 1000th term given a11 = 52 and a19 = 92. Explicit an = a1 + (n-1)d or an = a1 - d +dn a11 = a1 - d + 11d a19 = a1 - d + 19d 52 = a1 - d + 11d 92 = a1 - d + 19d 52 = a1 + 10d 92 = a1 + 18d -52 = -a d 92 = a1 + 18d 40 = 8d so d = 5 52 = a1 + 10(5) and a1 = 2 Therefore, an = n or an = 5n - 3 And finally A1000 = 5(1000) - 3 = 4997 Partial Sum (for n terms) Sn = ½n (a1 + an)

Geometric Sequence a1, a1 r, a1 r2, a1 r3, a1 r4,... where, a1 = first term and r = common ratio Recursive an=an-1r Explicit an = a1 rn-1 or an = (a1/r) rn Partial Sum (for n terms) Sn = a1(1 - rn)/(1 - r)

Geometric Example: Find the recursive and explicit formulas for the sequence {2, -10, 50, -250, 1250,... } r = -5, therefore an = 2(-5)n-1 or an = (2/-5)(-5)n which are equivalent Recursive an=an-1r Explicit an = a1 rn-1 or an = (a1/r) rn r = -5, therefore an = -5an-1

Geometric Example: Find the eight term for the sequence {5, 15, 45,... } r = 3, therefore an = 5(3)n-1 or an = (5/3)(3)n which are equivalent Recursive an=an-1r Explicit an = a1 rn-1 or an = (a1/r) rn r = 3, therefore an = 3an-1 and a8 = (5)(3)8-1 = 5(37) = 5(2187) = 10,935

Geometric Example: Find the sum of the first five terms of {1, 0.7, 0.49, } r = 0.7, therefore an = 1(0.7)n-1 Recursive an=an-1r Explicit an = a1 rn-1 or an = (a1/r) rn r =0.7, therefore an = 0.7an-1 and a5 = (1)(0.7)5-1 = 1(0.74) = Partial Sum (for n terms) Sn = a1(1 - rn)/(1 - r) and S5 = (1)( )/( ) = =2.7731

Geometric Example: Find a5 of a geometric sequence given a3 = 63/4 and a6 = 1701/32 a3 = a1r3-1 a6 = a1r6-1 63/4 = a1r2 1701/32 = a1r5 Solve by dividing: (a1r5/a1r2)=(1701/32)/(63/4) r = 3/2 a1 = 7 a5 = 7(3/2)4 = 567/16 Explicit an = a1 rn-1 or an = (a1/r) rn

Application: An amphitheater has 50 rows of seats with 30 seats in the first row, 32 in the second, 34 in the third, and so on. Find the total number of seats.

Application: As I was going to St. Ives I met a man with seven wives' Every wife had seven sacks; Every sack had seven cats; Every cat had seven kits; Kits, cats, sacks, and wives, How many were going to St. Ives? If the whole group goes (excluding observer), find the sum via partial sums methods!