9.1-9.3 Sequences and Series

As a special case, zero factorial is defined as

0! = 1

Arithmetic Sequence

A sequence in which each term is found by adding a fixed amount to the previous term

Geometric Sequence

A sequence in which each term is found by multiplying the previous term by the same number. Each pair of consecutive numbers have the same ratio

When you approach a question like, "Given two terms in a geometric sequence find the 8th term and the recursive formula." What do you do?

Divide bigger An by the smaller An and then root the the quotient to the root of bigger An- smaller An to find r. You can then use r to work your way back to A1 and set up the explicit formula

Initial Term

It is always assumed to be a1 , unless the problem specifically states otherwise

A geometric sequence is determined by:

Its initial term and its common ratio.

Can arithmetic sequences converge?

No infinite arithmetic sequence (such as 2 + 5 + 8 + 11+...) can have a sum, unless you include 0 + 0 + 0 + ... as an arithmetic sequence.

The sum of a convergent infinite geometric series with initial term a1 and common ratio r , where −1 < r < 1, lrl<1 is given by:

S = a1/1− r

Remember that Sn for a sequence starting with a1 is given by:

Sn = n ∑ak = a1 + a2 +...+ an k=1

The nth partial sum of an arithmetic sequence with initial term a1 and common difference d is given by:

Sn = n(a1 + an/2)

The nth partial sum of a geometric sequence with initial term a1 and common ratio r (where r ≠ 1) is given by

Sn= a1(1-r^n/1-r) Also could be written with a1 distributed to numerator

An infinite series converges (i.e., has a sum)

The Sn partial sums approach a real number (as n → ∞), which is then called the sum of the series if lim n→∞ Sn = S , where S is a real number, then S is the sum of the series

Cases where infinite series dont have a sum

The geometric series 2 + 6 + 18 + 54 + ... has no sum, because: n→∞ Sn = ∞ The geometric series 1− 1+ 1− 1+ ... has no sum, because the partial sums do not approach a single real number.

Common Difference

This is denoted by d . It is the number that is always added to a previous term to obtain the following term.

Common Ratio

This is denoted by r. It is the number that we always multiply the previous term by to obtain the following term. r = a2/a1= a3/a2

infinite sequence

a function whose domain is the set of positive integers. The function values a1, a2, a3, a4, . . . , an, . . .

finite sequence

a sequence whose domain consists of the first n positive integers only

In general, a recursive definition for an arithmetic sequence that begins with a1 may be given by:

a1 given an +1 = an + d (n ≥ 1; "k is an integer" is implied) (n+1 & n are subscripts)

In general, a recursive definition for a geometric sequence that begins with a1 may be given by:

a1 given an= an-1 ⋅r (n ≥ 1; "n is an integer" is implied) We assume a1 ≠ 0 and r ≠ 0 .

The general nth term of an arithmetic sequence with initial term a1 and common difference d is given by:

an = a1 + (n − 1) d Tip: Graph is Linear

FORMULA FOR THE GENERAL nth TERM OF A GEOMETRIC SEQUENCE

an = a1 ⋅r^(n−1) begin with a1 and keep multiplying by r until we obtain an expression for an Tip: Graph is Exponential

To find the first three terms of a sequence, given an expression for its nth term,

evaluate the expression for the nth term at n = 1 to find the first term, at n = 2 for the second term, and so on.

An arithmetic sequence is determined by:

its initial term and its common difference

If n is a positive integer, n factorial is defined as

n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n − 1) ⋅ n

How does an infinite geometric series converge?

⇔ (−1 < r < 1) i.e., r <1