Section 11.1 An Introduction to Sequences and Series

Sequences A sequence is a set of numbers or objects arranged in a particular pattern. Sequences may be finite or infinite. Sequences may be arithmetic, geometric or more complex. Sequences may converge or diverge. Each object in a sequence is called a term. Terms are numbered a1, a2, a3 etc. Example: The Fibonacci Sequence and its terms 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ……. a1 a5 a8 a13 

Using Algebra on Sequences For simplicity a sequence may be described algebraically. To find the nth term of a sequence, perform the appropriate substitution and solve the algebraic equation. Example: find the first 5 terms of an=2n+4 an=2n+4 a1=2(1)+4=6 a2=2(2)+4=8 a3=2(3)+4=10 a4=2(4)+4=12 a5=2(5)+4=14 We will revisit this data when we discuss summation notation. 

Recursive Sequences Sometimes a sequence requires knowledge of a previous term in order to be solved. This is called a recursive sequence. Example: find the next 3 terms of an=3(an-1)+2 if a1=3 an=3(an-1)+2 a1=3 a2=3(a2-1)+2 =3(a1)+2 =3(3)+2=11 a3=3(a2)+2=3(11)+2=35 a4=3(a3)+2=3(35)+2=107 

Writing rules for sequences Write a rule for the given sequence. 1, 3, 5, 7, …. Find the 200th term of the sequence.

Writing rules for sequences Write a rule for the given sequence. Find the 35th term of the sequence.

Summation Notation Sometimes we are interested in a series which is the sum of several terms of a sequence. The sum of a portion of a sequence is written using the sigma symbol (). Example: find the sum of the first 5 terms of an=2n+4 This can also be written as an=2n+4 a1=2(1)+4=6 a2=2(2)+4=8 a3=2(3)+4=10 a4=2(4)+4=12 a5=2(5)+4=14 

Section 11.2 Arithmetic Sequences

FlippedMath Users Sequences and Series can be found in the PreCalc tab (sections 14.1 and 14.2) 

KNOW THIS  Arithmetic Sequences An arithmetic sequence is a sequence where the difference between any two terms is a constant (d). We call this the common difference of the sequence. For each successive term of the sequence add d to the previous term. The general term of an arithmetic sequence is written as KNOW THIS 

Example

The Sum of the First n Terms of an Arithmetic Sequence The sum of the first n terms of an arithmetic sequence can be determined by the equation below. n is the number of terms to be summed a1 is the first term an is the nth term KNOW THIS 

Section 11.3 Geometric Sequences and Series

KNOW THIS KNOW THIS  Geometric Sequences A geometric sequence is a sequence where each successive term is obtained by multiplying the previous term by a nonzero constant (r). We call this ratio the common ratio of the sequence. The common ratio can be obtained from the following: KNOW THIS The general term of a geometric sequence is written as KNOW THIS 

Examples Geometric Sequence Common Ratio

Example

Example

The Sum of the First n Terms of a Geometric Sequence The series consisting of the sum of the first n terms of a geometric sequence can be determined by the equation below. n is the number of terms to be summed a1 is the first term r is the constant ratio, but r1 KNOW THIS 

Example

Example

Section 11.2 and 11.3 Word Problems

The Steps in Word Problems Figure out if the series is arithmetic or geometric Find the correct equation(s) (Recipe) Extract data from statement (Groceries) Plug the data in the equation(s) (Cook) Write the correct answer with units (Eat dinner) *you must show calculations and put the final answer in sentence form, WITH UNITS!

Example Over her 30 year career the teacher would earn at total of $1,189,385.39.

 The Value of an Annuity Annuities are common tools to build income for retirement. The value of an annuity (A) with the interest (r) calculated n times per year (t) can be obtained from the following formula: n is the number of times per year that deposits are made P is the amount deposited each time r annual interest rate t is the number of years deposits are made 

Example After 20 years you would have $74,445.20 in the account. Using the 4% rule you could spend $2997.81 every year without depleting the account.

Example After 30 years you would have $365,991.30 in the account. Using the 4% rule you could spend $14,639.65 every year without depleting the account.

Section 11.4 Infinite Geometric Series

Convergence, Divergence and the Sum of an Infinite Geometric Sequence When |r| <1 the sum (S) of an infinite geometric series may converge to a constant value using the following formula: a1 is the first term r is the constant ratio such that When |r| >1 of the infinite geometric does not have a sum and the series diverges. KNOW THIS 

The 5 main Equations Arithmetic Geometric KNOW THESE 

Example

Example Each $3,000 rebate check would add $10,000 to the economy.

Extra Material

Technology

Technology

Technology

Technology

Technology