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College Algebra
Pencil and Paper Homework
Name____________
P&P#6: Some Types of Sequences
Some Preliminary Definitions:
An arithmetic sequence (or an arithmetic progression) is an ordered list of numbers where the
difference of consecutive terms is always the same. This value is called the common difference (d).
Example: (1, 4, 7, 10, 13, …) is an arithmetic sequence with d =3.
A geometric sequence is an ordered list of numbers where the ratio (r) of consecutive terms is
always the same.
Example: (2, 6, 18, 54, …) is a geometric sequence with r =3.
A closed form for a sequence is a formula for computing any term of the sequence a n that doesn’t
depend on any of the other terms in the sequence. A closed form gives the sequence as a function of n.
1) Find the 0th, 1st, 2nd, 5th, and 21st terms of each of the following closed form sequences.
a)
a n  5n  11
b)
bn  3( 2) n 1
c)
cn 
d)
d n  3(1) n  7 (one might call this the “military sequence”)
5n  1
2n  1
2) Label each of the above sequences as arithmetic, geometric, or neither.
3) For what value of n will the sequence (a n ) first exceed 1,000,000? When will the sequence
(bn ) exceed first exceed 1,000,000?
College Algebra
Pencil and Paper Homework
Name____________
Definitions
A recursive form for a sequence is a formula for a n 1 that depends on some combination of the
previous terms a n , a n 1 , a n  2 , …
Recursive sequences need seed terms, which are beginning terms that allow you to build the
sequence.
Example: Suppose a sequence satisfies a n 1  2a n  a n 1 where a 0  1 and a1  3 . Some
people find it helpful to think about recursive sequences in terms of next, current,
and previous terms.
next term  2(current term)  ( previous term)
Since we know the 0th and 1st terms for the a n sequence, the next term is:
a 2  2a1  a 0  23  (1) .
4) Build the first several terms of the example recursive sequence:
a)
a 0 =1 a1 =3 a 2 =
a3 =
a4 =
a5 =
a6 =
5) Build the first 6 terms for each of the following sequences.
a)
a n 1  a n  3 (where a 0  7 )
b)
x n 1  3x n  x n 1 (where x 0  7 and x1  2 )
6) Write a recursive form for a sequence that requires 3 seed terms. Then, build the first 6 terms
of your sequence.
Triangular Numbers.
The sequence of triangular numbers T1 , T2 , T3 , … keeps track of the number of dots in triangular
arrays of increasing size.
…
T1  1
T2  3
T3  6
7) Sketch the triangular array for n= 4, 5, 6, and 7 above. Then, try to find a closed formula for
the nth triangular number T n .