Sequences and Series IB standard Students should know
![Sequences and Series IB standard • Students should know Arithmetic sequence and series; sum Sequences and Series IB standard • Students should know Arithmetic sequence and series; sum](https://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-1.jpg)
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 Sequence](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-2.jpg)
Arithmetic Sequence
![Arithmetic Sequences • An arithmetic sequence is a sequence in which each term differs Arithmetic Sequences • An arithmetic sequence is a sequence in which each term differs](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-3.jpg)
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 Algebraic Definition • {an} is arithmetic an+1 – an= d for all positive integers](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-4.jpg)
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 • a 1 is the 1 st term of an arithmetic The General Formula • a 1 is the 1 st term of an arithmetic](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-5.jpg)
The General Formula • a 1 is the 1 st term of an arithmetic sequence and the common difference is d • Then a 2 = a 1 + d therefore a 3 = a 1 + 2 d therefore a 4 = a 1 + 3 d etc. • Then an = a 1 + (n-1)d the coefficient of d is one less than the subscript
![No common difference! Arithmetic Sequence No common difference! Arithmetic Sequence](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-6.jpg)
No common difference! Arithmetic Sequence
![Example #1 • Consider the sequence 2, 9, 16, 23, 30… – Show that Example #1 • Consider the sequence 2, 9, 16, 23, 30… – Show that](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-7.jpg)
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 100 th term of the sequence – Is 828, 2341 a member of the sequence?
![Geometric Sequence Geometric Sequence](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-8.jpg)
Geometric Sequence
![Arithmetic Sequences Geometric Sequences • ADD To get next term • MULTIPLY to get Arithmetic Sequences Geometric Sequences • ADD To get next term • MULTIPLY to get](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-9.jpg)
Arithmetic Sequences Geometric Sequences • ADD To get next term • MULTIPLY to get next term • Have a common difference • Have a common ratio
![In a geometric sequence, the ratio of any term to the previous term is In a geometric sequence, the ratio of any term to the previous term is](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-10.jpg)
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!
![No common ratio! Geometric Sequence No common ratio! Geometric Sequence](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-11.jpg)
No common ratio! Geometric Sequence
![To write a rule for the nth term of a geometric sequence, use the To write a rule for the nth term of a geometric sequence, use the](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-12.jpg)
To write a rule for the nth term of a geometric sequence, use the formula:
![Write a rule for the nth term of the sequence 6, 24, 96, 384, Write a rule for the nth term of the sequence 6, 24, 96, 384,](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-13.jpg)
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.
![Write a rule for the nth term of the sequence 1, 6, 36, 216, Write a rule for the nth term of the sequence 1, 6, 36, 216,](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-14.jpg)
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.
![Write a rule for the nth term of the sequence 7, 14, 28, 56, Write a rule for the nth term of the sequence 7, 14, 28, 56,](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-15.jpg)
Write a rule for the nth term of the sequence 7, 14, 28, 56, 128, . . Then find
![One term of a geometric sequence is = 3. Write a rule for the One term of a geometric sequence is = 3. Write a rule for the](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-16.jpg)
One term of a geometric sequence is = 3. Write a rule for the nth term. The common ratio is r
![One term of a geometric sequence is and one term is Step 1: Find One term of a geometric sequence is and one term is Step 1: Find](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-17.jpg)
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 equation. Then, solve for. into your Step 3: Write the equation using r and .
![Write the rule when and . Write the rule when and .](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-18.jpg)
Write the rule when and .
![Series and Sequences Formulas Series and Sequences Formulas](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-19.jpg)
Series and Sequences Formulas
![Let’s graph the sequence we just did. Create a table of values. What kind Let’s graph the sequence we just did. Create a table of values. What kind](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-20.jpg)
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 Compound Interest Formula • P dollars invested at an annual rate r, compounded n](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-21.jpg)
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 Compound Interest • So if we invested $5000 that was compounded quarterly, at the](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-22.jpg)
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 Series and Sequences Formulas](http://slidetodoc.com/presentation_image_h/19c766f4a8a62b380ebfcb9278723458/image-23.jpg)
Series and Sequences Formulas
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