Sequences and Series IB standard Students should know Arithmetic sequence and series; sum of finite arithmetic series; geometric sequences and series; sum of finite geometric series

Arithmetic Sequence

Arithmetic Sequences An arithmetic sequence is a sequence in which each term differs from the pervious one by the same fixed number Example 2,5,8,11,14 5-2=8-5=11-8=14-11 etc 31,27,23,19 27-31=23-27=19-23 etc

Algebraic Definition {an} is arithmetic  an+1 – an= d for all positive integers n where d is a constant (the common difference)  “If and only if” {an} is arithmetic then an+1 – an is a constant and if an+1 – an is constant the {an} is arithmetic

The General Formula a1 is the 1st term of an arithmetic sequence and the common difference is d Then a2 = a1 + d therefore a3 = a1 + 2d therefore a4 = a1 + 3d etc. Then an = a1 + (n-1)d the coefficient of d is one less than the subscript

Find d for the following sequences 87, 83, 79, 75... d=-4 8, 9.5, 11, 12.5... d=1.5 19, 25, 31, 37... r=6 No common difference! Arithmetic Sequence 31, 37, 41, 47...

Example #1 Consider the sequence 2,9,16,23,30… Show that the sequence is arithmetic Find the formula for the general term Un Find the 100th term of the sequence Is 828, 2341 a member of the sequence?

Geometric Sequence

Have a common difference MULTIPLY to get next term Arithmetic Sequences Geometric Sequences ADD To get next term Have a common difference MULTIPLY to get next term Have a common ratio

Try to think of some geometric sequences on your own! In a geometric sequence, the ratio of any term to the previous term is constant. You keep multiplying by the SAME number each time to get the sequence. This same number is called the common ratio and is denoted by r What is the difference between an arithmetic sequence and a geometric sequence? Try to think of some geometric sequences on your own!

Find r for the following sequences 4, 8, 16, 32... r=2 r=3 8, 24, 72, 216... r=4 6, 24, 96, 384... No common ratio! Geometric Sequence 5, 10, 15, 20...

Writing a rule To write a rule for the nth term of a geometric sequence, use the formula:

Writing a rule Write a rule for the nth term of the sequence 6, 24, 96, 384, . . .. Then find This is the general rule. It’s a formula to use to find any term of this sequence. To find , plug 7 in for n.

Writing a rule Write a rule for the nth term of the sequence 1, 6, 36, 216, 1296, . . .. Then find This is the general rule. It’s a formula to use to find any term of this sequence. To find , plug 8 in for n.

Writing a rule Write a rule for the nth term of the sequence 7, 14, 28, 56, 128, . . .. Then find

(when you're not given the first term) Writing a rule (when you're not given the first term) One term of a geometric sequence is The common ratio is r = 3. Write a rule for the nth term.

(when you're given two non-consecutive terms) Writing a rule (when you're given two non-consecutive terms) One term of a geometric sequence is and one term is Step 1: Find r -divide BIG small -find the distance between the two terms and take that root. Step 2: Find . Plug r, n, and into your equation. Then, solve for . Step 3: Write the equation using r and .

(when you're given two non-consecutive terms) Writing a rule (when you're given two non-consecutive terms) Write the rule when and .

Series and Sequences Formulas

Graphing the sequence Let’s graph the sequence we just did. Create a table of values. What kind of function is this? What is a? What is b? Why do we pick all positive whole numbers? Domain, Input, X Range, Output, Y

Compound Interest Formula P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years. Think of P as the present value, and F as the future value of the deposit.

Compound Interest So if we invested $5000 that was compounded quarterly, at the end of a year looks like: After 10 years, we have:

Series and Sequences Formulas