Tuesday, June 2, 2009

Sequences & Series

A sequence is a function whose domain is only natural numbers.

In the general formula, it is common to use n instead of x.

  • Sequences can be infinite, so
    Sometimes, an extra term is needed, so
    precedes

Summation Notation:
n=> upper limit


i=> index of summation

1=>lower limit

Geometric Progressions or Geometric Sequences:
  • consecutive terms have a common ratio (r)

Methods of Representation:

1. Sequence: (list of numbers)
2. n-th term: (formulas)
3. Recursion: (using past terms)

Practice Word Problem:
A man wishes to save money by setting aside one cent the 1st, two cents the 2nd day, four cents the 3rd day and so on, doubling the amount each day. Assuming he does not run out of money, what is the total amount of money he would have saved at the end of 30 days?


  • First, you would recognize this as a geometric sequence. Using the equation, , you would plug in to solve for r.

  • Then with the given r value, which should be calculated as 2, you would then use the Geometric Sum formula, to find what the sum of the money would be.
  • With correct substitution of values, the answer comes out to be $3.00



How to find sequence/sum of series on calculators:
LIST MENU (2nd STAT) =>OPS=>seq(formula, variable, lower, upper)=>MATH sum(seq(...))


FORMULAS:
General Sequence:

General Series:

Arithemic Sequence:
nth term of an arithemic sequence:

Finding d:
Arithmetic Sum:
Geometric Sequence: Recursive Definition:

nth term of a geometric sequence:

Finding r:

Geometric Sum:

Infinite Series:




Practice Problems:

1. Find a formula for the nth term of the sequence below.

2. Find the 15th term of the geometric progression below.
2, 6, 18, 54,...

3. How many terms are in the sequence below?
4,12,36,108,...708588

4. The sum of the first n positive odd integers is 1521. Find n.

5. Insert 3 arithmetic means between 19 and 32.

6. The 7th and 10th terms of a geometric sequence are 821 and 6568. Find the 11th term.

7. The seats in a lecture room increase in each row from front to back. The first row has 10 seats. The second row has 11 seats, etc. Find the number of seats in the twelfth row.

8. Using the given information in problem 7, find the number of seats in the room if the twelfth row is the last row. 9. How many terms are in the sequence

10. For the sequence below, determine the next two terms:

1,5,13,29,61,...






Solutions to practice problems

1.

2. 9,565,938

3. 12 terms

4. n=39

5. 22.25, 25.5, 28.75

6. 13,136

7. 21 seats

8. 186 seats

9. 24 terms

10. 125, 253

2 comments:

Kate Nowak said...

Hi Mary and Kathryn -

Great job using Latex for the equations. They look beautiful.

Not all sequences are infinite! WHEN sequences are infinite, n approaches infinity.

Your post will be easier to read if your section titles are larger and bold.

It might be helpful to see a worked out example for at least some of the formulas, and/or an example of a word problem.

Kate Nowak said...

I'd like to see more of a discussion of the meaning of arithmetic sequences, like you did for geometric sequences.

Also, the answer to your word problem is incorrect. It's WAY more than $3.