CHAPTER 9A AN INTRO TO SEQUENCES

By Sheldon, Megan, Jimmy, and Grant.

9.1A

Sequence- list of numbers that usually form a pattern. Each number in the list is called a term.

Finite sequence 2,4,6,8 Infinite sequence 2,4,6,8……

9.1A

General rule an=2n where n is the # and an is the nth

term

The general rule can also be written in function notation: F(n)=2n

9.1A

Recursive sequence Must give you a1 or a1 and a2

Must give a rule for finding terms based on previous terms.

Example: Ak+1 = (-2)ak

9.1B

Factorial If n is a positive integer, then n!=n(n-1)(n-

2)… Example:

4! 4(3)(2)(1)= 24 Series

The sum of the terms in a sequence Can be finite or infinite

9.1B

Summation Notation Also called sigma notation(meaning Sum ∑

in Greek) Example: (i) is called the index of summation 5 is called the upper limit 1 is called lower limit

5

1

2i

9.1B

Summation notation for an infinite series

Example 2+4+6+8+10… would be

1

2i

9.2

Arithmetic Sequence Has a common difference between

consecutive terms That’s the number you add to each term to

get the next term Subtract any term by its previous

Rule for arithmetic sequence An=a1+(n-1)d

9.2

Sum of finite Arithmetic Sequence Sn=n/2 (a1 + an) Example:

2,8,14,20…n=25 S25= 25/2 (2+146) a25=2+(25-1)

(6) S25= 1850 a25= 146

9.3

Geometric Sequences Ratios of consecutive terms are the same

Example: (a2/a1)= r, a3/a2=r, a4/a3=r

To find the nth term of a geometric sequence you use (an)=a1r^(n-1)

9.3

To find the Sum of a Finite Geometric Sequence you use the formula

n

i

nir

r

raa

1

^1)1(^11

1

9.4

Mathematical Induction A mathematical proof about statements

involving positive integers Finite Difference

If all the first difference in the sequence are equal, then the sequence has a linear model an=an+b

If all the 1st differences are different, but the 2nd differences are equal, the sequence has a quadratic model an=an^2+bn+c