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Math 603
Remembering Sequences . . . .
Name ________________________
Instructions:
1. Write out the next three terms of each sequence given below.
2. Identify the type of sequence if you can. (arithmetic, geometric, etc.)
3. Write both an explicit and a recursive definition of each sequence where it is possible. (Note: some sequences
can be written as one, the other, or both. Some cannot be written as either.
Example:
Explicit
Recursive
t1  2

t n  t n 1 1
tn  n 1
2, 3, 4, 5 , 6 , 7

1. 140, 149, 158, ______, ______, ______
2. 15, 45, 135, ______, ______, ______
3. 25, 16, 9, 4, ______, ______, ______
4. 15, 11, 7, ______, ______, ______
5. 1, 3, 6, 10, ______, ______, ______
6. 72, 24, 8, ______, ______, ______
7. 1, 1, 2, 3, 5, 8, ______, ______, ______
8. 3, 8, 15, 24, ______, ______, ______
9. -10, 20, -40, ______, ______, ______
10. 9, 4, -1, ______, ______, ______
11. 4, 1, 0, 1, 4, 9, ______, ______, ______

METHOD OF FINITE DIFFERENCES
Given a sequence of integers, a1, a2 , a3, , we can
find a polynomial which explicitly generates any
term of the sequence by using the concept of finite
difference.
EXAMPLE: Find the polynomial which generates
the sequence: 4, 16, 44, 94, 172, . . . . .
4
16
+12

List several terms of the given sequence. Subtract
each term from the term after it to determine the
differences between the terms. Repeat this process
successively until a constant difference “k” is found.
(The number of subtractions determines the degree
of the polynomial.)
44
94
172
+28
+50
+78
+16
+22
+28
+6
+6
Note: It took 3
subtractions to obtain
the constant difference
6. So m =3 and k=6.
6n 3 6n 3
an 

 n3
3!
6!
This gives a sequence: 1, 8, 27, 64, 125, . . . .
knm
The first term of the polynomial will be
where

m!
m is the number of subtractions needed to reach the
constant difference k.
We now need to determine the
remaining terms of
the polynomial.
knm
For the sequence generated by this term (
) and
m!
subtract each of these terms from the corresponding
term of the original sequence. What is left will
determine the rest of the polynomial.
-
3
+5
The term you get (
knm
) will be the second term of
m!
the polynomial.
8
17
+9
+4
3
2
1
-
1
knm
For the 
sequence generated by this term (
) and
m!
16
8
8
44
27
17
30
+4
47
+17
Note: It took 2
subtractions to obtain
the constant difference
4. So m =2 and k=4.
17
18
-1
-1
-1
30
32
-2
-2
-1
1n1
 n
1!
-3
Note: It took 1
subtractions to obtain
the constant difference
-1. So m =1 and k=-1.
Next term:
Repeat steps 1 and 2 (ab0ve) on this new sequence.
Put together, we have the polynomial:
knm
The term you get (
) will be the Third term of
m!
the polynomial.
Continue this process until the linear term of the
polynomial
 (the one with power of 1) is obtained. 
You now have the polynomial with the exception of
a constant which must be added or subtracted to get
the desired sequence.
47
50
-3
-1
subtract each of these terms from the corresponding
term of the previous sequence. What is left will
determine the rest of the polynomial.

172
125
47
+4
8
8
0
0
-1
94
64
30
+13
4n 2
Next term:
 2n 2
2!

Repeat steps 1 and 2 (ab0ve) on this new sequence.
4
1
3

an  n 3  2n 2  n
This would give the sequence 2, 14, 42, 92, . . .
We need to add 2.
Final result: an  n 3  2n 2  n  2

SEQUENCES
Name __________________________
The following represent sequences which are not necessarily arithmetic or geometric. Write the explicit definition
of each sequence. Write the recursive definition also, if you see it.
1.
4, 9, 16, 25, 36,
5.
4, 1, 0, 1, 4, 9, 16, 25,
2.
5, 9, 19, 35, 57, 85,
6.
5, 17, 43, 89, 161, 265,
3.
0, -2, -8, -18, -32, -50,
7.
3, 5, 9, 17, 33, 65,
π.
O, T, T, F, F,
8.
-3, 6, 33, 90, 189, 342,
4.
4, 16, 54, 130, 256, 444,
t1  5
Figure out how to use the “Seq” MODE on your calculator to graph this sequence: 
2
t n  ( t n 1  2)
HINT: u(nMin) is the first term.
“u” is a standard letter (near the 7) and “n” is from the “X, T,  , n” key


Math 603
Review of Series
Name _________________________
Finding the value of a Series
Identify the sums below as either arithmetic or geometric and then find the actual value of the sum using the
methods from class (do not plug the numbers in your calculator and add.)
1. The sum of the first 100 positive integers.
2. The sum of the first 20 positive powers of 2.
3. The sum of the first 50 even positive integers..
4. The first 200 terms of the sequence: 3, 5, 7, 9, …
15
5. The series defined by
 60(.6)
n
n1
198
6. The series defined by

 (4n  60)
n 5
7. The series defined by: 10 + 14 + 18 + 22 + . . . + 214
8. The series defined by: .2 + .6 + 1.8 + 5.4 + . . . + 145.8

Expressing a sum using
9.

notation.
Express each of the series in 1-4 and 7-8 above using Sigma notation.
10. Express the first 250 terms of the sequence defined by 3, 4, 9, 18, 31, 48, . . .

(The hardest part of this one is finding the sequence to go with the numbers given – use all of your
knowledge of sequences to figure this one out. You may have to look back at your review notes on
sequences. Do not attempt to find the actual numerical value of the sum.
Some word problems
Set up the problems below as either geometric or arithmetic series, then find the sum using appropriate methods
from class.
11. A skyscraper is designed so that it slants in as the building increases in height. On the first floor of the
building there are 350 windows. On the next floor there are 348 windows. The number of windows
decreases by 2 on each successive floor. If the building is 100-stories (floors) tall, then how many windows
in total are on the building?
12. A checkerboard (or chess board) consists of 64 squares. A story states that when the King of Persia was first
introduced to the game of chess, he was so impressed that he decided he would reward the game’s inventor.
He asked the inventor what he would like as a reward. The inventor said he wanted grains of wheat
according ot the following scheme:
One grain on the first square of the chess board.
Two grains on the second square.
Four grains on the third square.
Eight grains on the fourth square.
Sixteen grains on the fifth square.
. . . . and son on for all 64 squares of the chessboard.
The king believed he was getting a good deal. Was he? Find the total grains of wheat the king had to give the
inventor.