Download 13.1 Arithmetic and Geometric Sequences

Arithmetic and
Geometric Sequences
Explicit Formulas

A "sequence" (or "progression", in British
English) is an ordered list of numbers; the
numbers in this ordered list are called
"elements" or "terms".


A sequence may be named or referred to as "A" or
"An". The terms of a sequence are usually named
something like "ai" or "an", with the subscripted letter
"i" or "n" being the "index" or counter. So the second
term of a sequence might be named "a2"
(pronounced "ay-sub-two"), and "a12" would
designate the twelfth term.
Note: Sometimes sequences start with an index of n
= 0, so the first term is actually a0. Then the second
term would be a1. The first listed term in such a case
would be called the "zero-eth" term. This method of
numbering the terms is used, for example, in
Javascript arrays. Don't assume that every sequence
and series will start with an index of n = 1.
Arithmetic Sequences
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
The two simplest sequences to work with are
arithmetic and geometric sequences. An
arithmetic sequence goes from one term to the
next by always adding (or subtracting) the same
value.
For instance, 2, 5, 8, 11, 14,... and 7, 3, –1, –5,...
are arithmetic, since you add 3 and subtract 4,
respectively, at each step.

The number added (or subtracted) at each
stage of an arithmetic sequence is called
the "common difference" d, because if you
subtract (find the difference of) successive
terms, you'll always get this common
value.
Find the common difference and
the next term of the following
sequence: 3, 11, 19, 27, 35,...
The difference is always 8, so d = 8.
Then the next term is 35 + 8 = 43.

For arithmetic sequences, the common
difference is d, and the first term a1 is
often referred to simply as "a". Since you
get the next term by adding the common
difference, the value of a2 is just a + d. The
third term is a3 = (a + d) + d = a + 2d. The
fourth term is a4 = (a + 2d) + d = a + 3d.
an = a1 + (n – 1)d
Examples: Find a formula for an
and find the 10th term

2,6,10,14,18,…

17,10,3,-4,-11,-18,…
Examples:

Find the n-th term (formula) and the
first three terms of the arithmetic
sequence having a4 = 93 and a8 = 65.
Since a4 and a8 are four places apart, then I
know from the definition of an arithmetic
sequence that a8 = a4 + 4d.
65 = 93 + 4d
–28 = 4d
–7 = d
93 = a + 3(–7)
93 + 21 = a
114 = a
OR
65 = a + 7(–7)
65 + 49 = a
114 = a
Solution
an=114+(n-1)(-7)
=114-7n+7
 an=121-7n
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
a1=114, a2=107, a3=100
Examples:

Find the n-th term (formula) and the
tenth term of the arithmetic sequence
having a2 = 2 and a5 = 16.
an=-8/3+(n-1)(14/3)
 an =14/3n-22/3


a10=118/3
Geometric Sequences
A geometric sequence goes from one term
to the next by always multiplying (or
dividing) by the same value.
 So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1,
1/3,... are geometric, since you multiply by
2 and divide by 3, respectively, at each
step

The number multiplied (or divided) at each
stage of a geometric sequence is called
the "common ratio" r, because if you divide
(find the ratio of) successive terms, you'll
always get this common value.
Find the common ratio and the
seventh term of the following
sequence:
2/9, 2/3, 2, 6, 18,...
The ratio is always 3, so r = 3.
Then the sixth term is (18)(3) = 54 and the
seventh term is (54)(3) =162
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For geometric sequences, the common
ratio is r, and the first term a1 is often
referred to simply as "a". Since you get the
next term by multiplying by the common
ratio, the value of a2 is just ar. The third
term is a3 = r(ar) = ar2. The fourth term is
a4 = r(ar2) = ar3.
an = a1
(n
–
1)
r
Examples: Find a formula for an
and find the 10th term

1,3,9,27,81,…

64,-32,16,-8,4,…
Example

Find the n-th (formula) and the 26th
term of the geometric sequence with
a5 = 5/4 and a12 = 160.
These two terms are 12 – 5 = 7 places apart, so, from the
definition of a geometric sequence, I know that
7
160 = (5/4)(r )
128 = r7
2=r
4
5/4 = a(2 ) = 16a
5/64 = a
a12=a5r7
11
160 = a(2 ) = 2048a
OR
160/2048 =5/64= a
Solution
an=5/64(2)(n-1)
 an=5/128(2)n
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
a26=2,621,440
Example

Find the n-th (formula) and the 11th
term of the geometric sequence with
a3 = 12 and a6 = 96.
an=3(2)(n-1)
 an=3/2(2)n

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a11=3072