PreCalculus Generic Notes
© by Scott Surgent
Sequences and Summation Notation
A sequence is a function whose domain is the positive integers (sometimes 0). Instead of
f (x ) notation, we use a n notation, where the n is the input variable, called the index,
and a n is the output result.
If a n is defined, then it is just a matter of plugging in values for n to determine a n . For
example, if a n = n 2 + 1 , then a 5 = 5 2 + 1 = 26 . The function a n is called the general nth
term of the sequence.
Determining a n from a list of numbers can be tricky, but this is where your pattern
recognition skills come in. View the examples in your text for ‘inspiration’.
We are introduced to n! , read as n-factorial. This is a common counting item defined as
the product of all integers from 1 to n (for example, 4! = 1 x 2 x 3 x 4 = 24). 0! Is defined
as 1. It appears quite a bit in sequences and series.
If we sum the members of a sequence we have a series. Please review your text for
examples on summation notation. We will explore series in a few sections.
Arithmetic Sequences
An arithmetic sequence is a sequence in which each term is found by adding a constant to
the preceding term. This constant is called the common difference. The plot of the points
of an arithmetic sequence is linear; hence we can use our linear algebra equations to help
us in this section.
The general nth term of an arithmetic sequence is a n = dn + c . Does this look similar to
a linear equation? The d is the common difference, akin to the slope from linear
equations. I’d suggest you simply form these arithmetic sequences by using your skills
from linear equations. Pick two points and use the slope-intercept formula. See below:
Suppose you are told that the 4th term of an arithmetic sequence is 20 and the 13th term is 65.
Convert the given information into two points (4,20) and (13,65). Use our slope formula to
find d: d = 6513−−204 = 459 = 5 . We now have a n = 5n + c . To find c, just plug in one of the
points, say (4,20) to get 20 = 5( 4) + c , so c = 0 . Hence the general nth term is a n = 5n .
There are two sum formulas for arithmetic series:
S = n2 (a1 + a n ) and S = n2 (a1 + (n − 1)d ) .
The n stands for the number of terms to be summed, a1 is the first term, and a n is the
last term. The d is the common difference. If you don’t know the last term, use the
PreCalculus Generic Notes
© by Scott Surgent
second formula. If you know the last term, use the first. Study the examples in your text.
These formulas are actually the same, just dressed a little differently.
Geometric Sequences
A geometric sequence is one in which each term is found by multiplying the preceding term
by a constant. This constant is the common ratio, and is denoted by the letter r. The
common ration can easily be found by taking any two consecutive terms of a geometric
sequence and dividing the second by the first. The general nth term of a geometric
sequence has the format a n = ar n −1 .
The sum formula for a geometric series is:
S=
a1 (r n − 1)
(r − 1)
This form is a little nicer since it usually involves fewer negative signs.
An interesting use for this sum formula involves infinite geometric sequences. If the
common ratio r is between –1 and 1, then the term r n will tend toward 0 as n tends to
infinity, leaving us with the infinite geometric sum formula S = 1−ar . Ever wonder what
1
2
+ 14 + 18 + 161 + L equals? This is geometric, with a =
S=
1
2
=
1
2
and r = 12 , so the infinite sum is
1
2
1
2
= 1 . This shouldn’t be surprising; imagine slicing a pizza in half, then in
1 − 12
half again, so forth. “1 pizza” = half a slice + a quarter slice + an eighth slice + so forth.
Mmmm, pizza.
Keep in mind this concept is a limit. Can you really add infinitely many terms? Probably
not. But (this is important) you can get arbitrarily close to 1 by going out enough terms.
The sum approaches 1 as a limit.
The Binomial Theorem
The binomial theorem is a shorthand for expanding binomials to various powers, i.e.,
problems like ( x + y ) n , where n is a positive integer. To start, we cite some easy ones:
( x + y) 0 = 1
( x + y )1 = x + y
( x + y ) 2 = x 2 + 2 xy + y 2
( x + y ) 3 = x 3 + 3 x 2 y + 3 xy 2 + y 3
PreCalculus Generic Notes
© by Scott Surgent
This sort of expansion by hand is tedious, but we detect some patterns: The x terms count
down from the starting n value, all the way to 0 (the last term contains an x 0 , which is
just 1). The y values count up from 0 (the first term contains a y 0 , which is 1) all the
way to n.
The coefficients follow an interesting pattern known as Pascal’s Triangle, found on page 736.
Incidentally, this pattern was known at least 2,000 years before Pascal’s lifetime.
Individually, these numbers are called binomial coefficients, and are defined by the value
C (n, m) , where n is the power of the binomial, hence the nth row of the triangle (the top 1 is
the 0th row), and m is the mth place of that row, from the left, with the first spot being the 0th
place. (Your calculator should have this key; on the TI-82/83, it’s under MATH/PRB,
written as n C r. First type the n, then find nCr, then type the r, and hit ENTER.)