Sequences and Mathematical Induction Chapter 4 Sequences and Mathematical Induction

4.1 Sequences

Sequences The main mathematical structure used to study repeated processes is the sequence. The main mathematical tool used to verify conjectures about patterns governing the arrangement of terms in sequences is mathematical induction.

Example Ancestor counting with a sequence two parents, four grandparents, eight great-grandparents, etc. Number of ancestors can be represented as 2position Example: 23 = 8 (great grandparents), therefore parents removed three generations are great grandparents for which you have a total of 8.

Sequences Sequence is a set of elements written in a row as illustrated on prior slide. (NOTE: a sequence can be written differently) Each element of the sequence is a term. Example am, am+1, am+2, am+3, …, an terms a sub m, a sub m+1, a sub m+2, etc. m is subscript of initial term n is subscript of final term

Example Finding terms of a sequence given explicit formulas ak = k/(k+1) for all integers k ≥ 1 bi = (i-1)/i for all integers i ≥ 2a the sequences a and b have the same terms and hence, are identical a1 = 1/(1+1) = ½ b2 = (2-1)/2 = ½ a2 = 2/(2+1) = 2/3 b3 = (3-1)/3 = 2/3 a3 = 3/(3+1) = 3/4 b4 = (4-1)/4 = 3/4 a4 = 4/5 b5 = 4/5

Example Alternating Sequence cj = (-1)j for all integers j≥0 sequence has bound values for the term. term ∈ {-1, 1} c0 = (-1)0 = 1 c1 = (-1)1 = -1 c2 = (-1)2 = 1 c3 = (-1)3 = -1 c4 = (-1)4 = 1 …

Example Find an explicit formula to fit given initial terms sequence = 1, -1/4, 1/9, -1/16, 1/25, -1/36, … What can we observe about this sequence? alternate in sign numerator is always 1 denominator is a square ak = ±1 / k2 (from the previous example we know how to create oscillating sign sequence, odd negative and even positive. ak = (-1)k+1 / k2 1/12 -1/22 1/32 -1/42 1/52 -1/62 a1 a2 a3 a4 a5 a6

Summation Notation Summation notation is used to create a compact form for summation sequences governed by a formula. the sequence is governed by k which has lower limit (1) and a upper limit of n. This sequence is finite because it is bounded on the lower and upper limits.

Example Computing summations a1 = -2, a2 = -1, a3 = 0, a4 = 1, and a5 =2.

Example Computing summation from sum form.

Example Changing from Summation Notation to Expanded form

Example Changing from expanded to summation form. Find a close form for the following:

Separating Off a Final Term A final term can be removed from the summation form as follows. Example of use: Rewrite the following separating the final term

Example Combining final term

Telescoping Sum Telescoping sum can be evaluated to a closed form. See book page 205

Product Notation Recursive form

Example Compute the following products:

Factorial Factorial is for each positive integer n, the quantity n factorial denoted n! is defined to be the product of all the integers from 1 to n: n! = n * (n-1) *…*3*2*1 Zero factorial denoted 0! is equal to 1.

Example Computing Factorials

Properties of Summations and Products Theorem 4.1.1 If am, am+1, am+2, … and bm, bm+1, bm+2, … are sequences of real numbers and c is any real number, then the following equations hold for any integer n≥m:

Examples Let ak = k +1 and bk = k – 1 for all integers k

Examples Let ak = k +1 and bk = k – 1 for all integers k

Transforming a Sum by Change of Variable Transform the following by changing the variable. summation: change of variable: j = k+1 Solution: compute the new limits: lower: j=k+1, j=0+1=1 upper: j=k+1, j=6+1=7