Algebra Accelerated
Recursive Sequences
In a recursive formula each term is defined as a function of its preceding term(s).
Each term is found by doing something to the previous term(s). A recursive formula
designates the starting term, a1 , and the n th term of the sequence, a n , as an
expression containing the previous term (the term before it), a n1 .
Examples:
1) Find the first three terms in the sequence an  3an1  4 if a1  5 .
2) Find the recursive formula for the following sequence:
2, 8, 14, 20, 26…
(1) a1  2 , an  an1  6
(3) a1  2 , an  4(an1 )
(2) a1  2 , an  2(an1 )  2
(4) a1  2 , an  an1  6
3) The diagrams below represent the first three terms of a sequence.
Assuming the pattern continues, which formula determines the a n , the number of shaded squares
in the n th term?
(1) a n  a n1  4
(2) an  an1  4
(3) an  an1  4
(4) an  an1  4
4) A sequence is defined recursively by f (1)  16 and f (n)  f (n  1)  2n . Find f (3).
(1) 32
(2) 30
(3) 28
(4) 26
Algebra Accelerated
Directions: Find the first 4 terms in each of the following sequences
5). an1  an  6 where a1  12
6). f (n  1)  2 f (n)  8 and f (1)  3
7). A sequence is given using a1  3 and an  an1  6 . Which of the following is the value of a 4 ?
(1) 24
(2) 12
(3) 21
(4) 15
Explicit Sequences
An explicit formula for a sequence is a formula in which any term in a sequence can
be determined. It defines the n th term of the sequence as an expression of n where
n is the “location” of the term. Any term can be determined by substituting n into the
formula.
Examples:
8) Find the 7th term in the sequence
an  2n  4
10) Find the first three terms in the sequence
an  n 2  1
9)
Find the 5th term of the sequence a n  3 n
11) Find the fifth term of the sequence
defined as: a(n)  (n  3) n1
Algebra Accelerated
12) Given a1  5 and an  an1  4 , find the explicit formula.
(1) an  n  4
(3) a n  4n  1
(2) a n  4n  1
(4) an  4n  4
13) The diagram below represents the first three terms of a sequence.
Assuming the pattern continues, which formula determines a n , the number of shaded squares in
the n th term?
(1) an  4n  12
(2) an  4n  8
(3) an  4n  4
(4) an  4n  2
Arithmetic Sequences
 Adding pattern or subtracting pattern
 Formula: an  a1  (n  1)d
a n : term desired
a1 : first term in the sequence
n : location in the sequence of the term desired
d : is the common difference
Examples:
14) Find the explicit formula for the sequence. 10, 2, -6, -14
(1) a n  8n  18
(3) an  8n  18
(2) an  n  26
(4) a n  8(n  18)
15) Find the 7th term of an arithmetic sequence if a1  5 and d  2 .
Algebra Accelerated
16) Determine whether the following sequences are arithmetic or geometric and explain your
reasoning. 5, 10, 15, 20, …
17) In an arithmetic sequence a1  5 . Find a10 if a6  17 and a7  19
18) Write the formula for this sequence {1,4,7,10,13...}
(a) Find the 6th term of this sequence
(b) Find the 20th term of this sequence
19) Which function could be used to generate the sequence 10, 15, 20, 25, 30, 35, …?
(1)
(2)
(3)
(4)
f (n)  10n  5
f ( n)  n  5
f (n)  10  5(n  1)
f (n)  5  5(n  1)
Geometric Sequences
 Multiplying pattern or dividing pattern
 Formula: a n  a1 r n 1
a n : term desired
a1 : first term in the sequence
n : location in the sequence of the term desired
r : is the common difference
Examples:
20) What is the 5th term of the sequence 5, 10, 20, 40,…
Algebra Accelerated
21) What is the common ratio of the geometric sequence -2, 4, -8, 16, …?
(1) 
1
2
(2) 2
(3) -2
(4) -6
22) A sequence has the following terms: a1  3 , a2  12 , a3  48 , a4  192 . Which formula
represents the nth term in the sequence?
(1) an  3  4n
(3) a n  3( 4) n
(2) an  3  4(n  1)
(4) a n  3(4) n 1
Converting Between Recursive and Explicit Formulas
Step 1: Use the explicit formula to determine the first 3 terms of the sequence
Step 2: List out the terms in order and determine the pattern
Step 3: Use the patter you discovered to write the recursive formula
23)
Algebra Accelerated
24)
25)
26)
27)
28) If f (1)  3 and f (n)  2 f (n  1)  1 , then f (5) 
(1) -5
(2) 11
(3) 21
(4) 43
Algebra Accelerated
29) If a sequence is defined recursively by f (0)  2 and f (n  1)  2 f (n)  3 for n  0 , then f (2) is
equal to
(1) 1
(2) -11
(3) 5
(4) 17
30) A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Which function(s) shown
below can be used to determine the height, f (n) , of the sunflower in n weeks?
I.
II.
III.
f ( n)  2n  3
f (n)  2n  3(n  1)
f (n)  f (n  1)  2 where f (0)  3
(1) I and II
(2) II, only
(3) III, only
(4) I and III
31) The diagram below represents the first three terms of a sequence.
Assuming the pattern continues, which formula determines a n , the number of shaded
squares in the n th term?
(3) an  4n  12
(4) an  4n  8
(3) an  4n  4
(4) an  4n  2
32) What is a formula for the nth term of sequence B shown below?
B = 10, 12, 14, 16, …
(1) bn  8  2n
(3) bn  10(2) n
(2) bn  10  2n
(4) bn  10(2) n 1
Algebra Accelerated
33) Find the first four terms of the recursive sequence defined below.
a1  3
a n  a( n 1)  n
(1) -3, -5, -7, and -9
(2) -3, -1, 2, and 5
(3) -3, -4,-5 and -6
(4) -3, -5, -8 and -12
34) A grocery store stacks soup cans to look like a triangle. There is one can at the top, followed by
2 can below that, then 3 cans, then 4 cans and so forth. How many cans would be used in a
display of 6 rows?
(1) 17
(2) 19
(3) 21
(4) 23
35) What is the formula for the nth term of the sequence 54, 18, 6, …?
(1)
(3)
(2)
(4)
36) Haley wishes to build a "tower" out of blocks so that each row has two more blocks than the
row above it, as shown in the drawing. If Haley begins with 60 blocks, how many complete rows
of this tower will she be able to build?
(1) 5
(2) 6
(3) 7
(4) 8
37) Find the explicit formula for the following sequence.
1, 3, 5, 7, 9
(1) a (n)  2n  1
(2) a(n)  n  2
(3) a (n)  2n  1
(4) a(n)  2n  2
Algebra Accelerated
38) Find the explicit formula for the following sequence.
−2, −3, −4, −5, −6
(1) a (n)  n  1
(2) a(n)  n  1
(3) a (n)  (n  1)
(4) a(n)  n  1
39) Find the recursive formula for the following sequence:
2, 8, 14, 20, 26
(1) a  2 , a n  a n 1  6
(3) a1  2 , a n  2a n 1  2
(2) a  2 , a n  4a n 1
(4) a1  2 , a n  a n 1  6
40) Find the explicit formula for the sequence defined by the following recursive formula:
a1  2 , a n  a n 1  3
(3) an  (n  3)  2(n  1)
(4) an  4n  5
(1) an  3n  5
(2) an  (n  3)  2(n  1)
41) Find the explicit formula for the sequence:
2, 5, 8, 11, 14
(1) a(n)  3n  1
(2) a (n)  2n  1
(3) a(n)  3(n  1)
(4) a(n)  3n  1
42) Find the explicit formula for the sequence:
10, 2, −6, −14
(1) a (n)  8n  18
(2) a(n)  n  26
(3) a(n)  8n  18
(4) a (n)  8(n  18)
43) Find the explicit formula for the following sequence:
−2, −6, −18, −54
(1) a n  2(3) n 1
(3) a n  2(3) n 1
(2) a n  2(3) n
(4) a n  2(3) n 1
Algebra Accelerated
44) A sequence has the following terms: a1  4 , a2  10 , a3  25 a4  62.5 . Which formula
represents the nth term in this sequence?
(1) a(n)  4  2.5n
(2) a(n)  4  2.5(n  1)
(3) a(n)  4  2.5 n
(4) a(n)  4  2.5 n1
45) The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a1,
which is an equation for the nth term of this sequence?
(1) a (n)  8n  10
(3) a (n)  16n  10
(2) a (n)  8n  14
(4) a (n)  16n  38
46) What is the common difference, d, in the arithmetic sequence defined by the formula
a ( n)  2n  1
(1) 1
(2) 2
(3) n
(4) 2n
47) Find the 22nd term of the arithmetic sequence with a1  6 and a common difference of d = 3.
(1) 54
(2) 60
(3) -63
(4) 57
48) Find the 40th term of the arithmetic sequence with a1  150 and a common difference of d = −4.
(1) -6
(2) 6
(3) -2
(4) 2
49) The 16th term of an arithmetic sequence is −8 and the first term is 12. What is the common
difference, d?
(1) -4
(2)
3
4
(3)
4
3
(4) 4