Math SATs- Sequences and Series
Welcome to a Math SAT Preparation Lesson. In this lesson we will study sequences and series.
Let's start off with sequences
A sequence is a list of numbers that has a certain order. For example, 16,7,888,-2,0,72 is a sequence, since 16 is the first number in the list, 7 is the second, etc. Sequences can have a certain "rule" by which terms progress, but they can also be completely random.
A few examples of cool sequences I can think of off the top of my head:
1,4,9,16… a sequence of the perfect squares starting from 1.
1,1,2,3,5,8,13,21,34,... the Fibonacci sequence, where each term is the sum of the previous two terms.
1, ,
,... the harmonic sequence.
0,0,0,0… the "empty" sequence.
There are 2 special types of sequences we like to look at- arithmetic ones and geometric ones.
Arithmetic sequences are sequences that
start with any number a, and in which every term can
be written as
,
where d is any number. An example of such a sequence would be 5,
12, 19, 26, 33…, where
and
. This is an increasing
arithmetic sequence, as the terms are increasing. Decreasing
arithmetic sequences have
.
Geometric sequences also start with any
number a (though usually a is nonzero here), but this time we are
not adding an extra d value each time- we multiply a by a factor
of r. Thus, the term is
. Geometric sequences can either be monotonic,
when r is positive and the terms are moving in one direction, or
alternating, where
and the terms
alternate between positive and negative values, depending on n.
Sequences are fun, but they have an important application- series.
A series is a sequence of numbers that represent partial sums for another sequence. For example, if my sequence is 1,2,3,4… then my series would be 1,1+2,1+2+3,..., or 1,3,6,10….
With arithmetic and geometric series, we can use a formula to
calculate any term of the
series. These shortcuts are useful since they save you from
having to write out the entire sequence and add all of its terms
up.
The formula for an arithmetic series is:
The formula for a geometric series is:
These formulas are worth memorizing.
The trick in solving sequence and series problems is recognizing first the type of sequence you're dealing with, and then finding the proper a, d or r if you're dealing with an arithmetic or geometric ones.
Are you a king of sequences?
Can you spot the patterns and calculate the term?
Check out the
Sequence and Series test.
Post Comments
oLahav said – Wed, 04 Jun 2008 13:27:54 -0000 ( Flag Edit Link )
The harmonic sequence just means that for each n from 1 to infinity, the nth term of the sequence is 1/n.
On the SATs you may be asked about any sort of sequence- arithmatic, geometric, even some random ones you’ll have to spot yourself. The formulas given here are most frequently asked about and they’ll also help you in your later studies of sequences and series in terms of calculus limits, if you go on to study math at college, so I thought including them is a good idea.
For any other questions, post a discussion. And make sure to check out the Sequences test to know exactly what type of questions you’ll be asked.