Sequences and Mathematical Induction An important task of

Sequences and Mathematical Induction An important task of mathematics is to discover and characterize regular patterns, such as those associated with repeated processes. • The main mathematical structure to study repeated processes is the sequence. • The main mathematical tool to verify conjectures about patterns in sequences is mathematical induction. 1

Sequences • Sequence is a set of elements written in a row: am, am+1, …, an. • The elements are called terms. • k is called subscript or index of ak. • am is the initial term; an is the final term. • am, am+1, am+2, … is an infinite sequence. 2

Sequences • Sequences characterize regular patterns. • Examples: 1) 1, 8, 15, 22, 29. 2) 2, 4, 8, 16, 32, 64, … 3) 2, 3, 5, 7, 11, 13, 17, … 3

Explicit Formula for a sequence • Explicit (general) formula is a rule that shows how the values of ak depend on k. • Examples: 1) ak=1+7 k for 1, 8, 15, 22, 29. 2) bk=2 k for 2, 4, 8, 16, … 3) ck= (-1)k · (2 k+1) for -3, 5, -7, 9, … 4

Summation Notation Ø Let m and n be integers such that m ≤ n. Then We call k index of the summation; m the lower limit of the summation; n the upper limit of the summation. Ø Ex. : Suppose a 3=2, a 4=-4, a 5=0, a 6=7. Then 5

Explicit formula for summation • Example: If ak=2 k then • Note that the index of summation is a dummy variable, so can be replaced by any other symbol: Ex: i=k+1 is called change of variable. 6

Product Notation Ø Let m and n be integers such that m ≤ n. Then Ø Examples: For each n Z+, is called n factorial. E. g. , 4! = 1 · 2 · 3 · 4 = 24 Note: 0! = 1 7

Binary representation of integers Ø Recall that if a = pk · 2 k + pk-1 · 2 k-1 + … + p 1 · 21 + p 0 · 20 then a 10 = (pk pk-1 … p 1 p 0 )2 , where p 0, p 1, …, pk-1, pk is a sequence of binary digits 0 and 1. Ø Question: How to find p 0, p 1, …, pk-1, pk ? 8

Converting from base 10 to base 2 14 = 7 · 2 + 0 7=3· 2+1 3=1· 2+1 1=0· 2+1 14 = 7 · 21 + 0 · 20 = ( 3· 2 + 1 ) · 21 + 0 · 20 = 3 · 2 2 + 1 · 21 + 0 · 2 0 = ( 1· 2 + 1 ) · 22 + 1 · 21 + 0 · 20 = 1 · 23 + 1 · 22 + 1 · 21 + 0 · 20 9

Converting from base 10 to base 2 Ø 14 = 1 · 23 + 1 · 22 + 1 · 21 + 0 · 20 Thus, 1410 = 11102 Ø Generally, to get binary representation for nonnegative integer a, Repeatedly divide by 2 until a quotient of zero is obtained. If the remainders found are r[0], r[1], …, r[k], then a 10 = ( r[k] r[k-1] … r[1] r[0] )2. 10