Download Algebra II/Trig Honors Unit 7 Day 1: Define and Use Sequences and

Algebra II/Trig Honors
Unit 7 Day 1: Define and Use Sequences and Series
Objective: To recognize and write rules for number patterns
Find the next two terms. Describe the pattern you continued in words.
-5, -2, 1, 4, _________, _________ .
1, 4, 9, 16, _________, _________ .
3, 9, 27, 81, _________, _________ .
Key Terms:

Sequence - ______________________________________________________________
________________________________________________________________________
o If the domain is not specified, _________________________________________.
o The values in the range are called the ___________________________________.
Domain: 1
2
3
4 . . . n
The position of each term
Range: a1
a2
a3
a4 . . . a n
Terms of the sequence

Finite sequence - _______________________________________________________
o Ex. 2, 4, 6, 8

Infinite sequence - ______________________________________________________
o Ex. 2, 4, 6, 8, …

A sequence can be specified by an equation, or rule. Both sequences above can be described by the rule
a n  2n or f n  2n .
Example 1: Write terms of sequences
Write the first six terms of each sequence.
a. an  2n  5
b. f n    3
n 1
Writing Rules - If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for
the n-th term of the sequence.
Example 2: Write rules for sequences
Describe the pattern, write the next term, and write a rule for the n-th term of the sequences.
a. -1, -8, -27, -64, . . . . .
b. 0, 2, 6, 12, . . . . .
Graphing Sequences – To graph a sequence, let the horizontal axis represent the position numbers (the
domain) and the vertical axis represent the terms (the range).
Example 3: Solve a multi-step problem
You work in a grocery store and are stacking apples in the shape of a square pyramid with 7 layers. Write a rule
for the number of apples in each layer. Then graph the sequence. (Let the top of the pyramid be the first layer).
More Key Terms:

Series - ________________________________________________________________

o Finite series: 2 + 4 + 6 + 8
o Infinite series: 2 + 4 + 6 + 8 + . . . .
Summation Notation - ___________________________________________________

4
Finite Series: 2 + 4 + 6 + 8 =
 2i
i 1
Infinite Series: 2 + 4 + 6 + 8 + . . . =
 2i
i 1
Read as “the sum of 2i for values of i from 1 to 4”
o The index of the summation is i (can be any letter – most common: i, k, and n)
o The lower limit of summation is 1
o The upper limit of summation is 4 for the finite series,  for the infinite series
Example 4: Write series using summation notation
Write the series using summation notation.
a. 25 + 50 + 75 + . . . + 250
b.
1 2 3 4
    ....
2 3 4 5
Example 5: Find the sum of a series
 3  k 
8
Find the sum of the series
2
k 4
Special Formulas for Special Series (note: The index MUST start at one in order to use the forms.)
n
1. Sum of n terms of a constant:
 c  cn
i 1
n
2. Sum of first n positive integers:
i 
i 1
nn  1
2
n
3. Sum of the squares of first n positive integers:
i
i 1
2

nn  12n  1
6
*Rewrite example 5 and use the formula to find the sum.
Example 6: Use a formula for a sum
How many apples are in the stack in Example 3? The rule for the number in each layer was tn = n 2 .
Class Practice:
Write the first six terms of the sequence.
1.
tn = n + 4
2.
an = (-2)
n-1
3.
f ( n) =
n
n +1
4. For the sequence 3, 8, 15, 24, … determine the pattern and write the next term. Graph the first five
terms and write a rule for the nth term.
Write the series using summation notation.
5.
5+10 +15+... +100
6.
1 4 9 16
+ + + +...
2 5 10 17
7.
6 + 36 + 216 +1296 +...
8. 5+ 6 + 7+... +12
Find the sum of the series.
5
9.
34
å8k
11.
k=1
i=1
7
10.
å1
6
å( k 2 -1)
12.
k=3
ån
n=1
HW: Page 438 #4-56 (M4), 58-62
Answers: 1. 5,6,7,8,9,10 2. 1,-2,4,-8,16,-32 3.
20
5.
å 5k 6.
k=1
¥
i2
å i 2 +1 7.
i=1
¥
å 6i 8.
i=1
1 2 3 4 5 6
4. Next term = 35, rule an = n ( n + 2)
, , , , ,
2 3 4 5 6 7
8
å( 4 + i)
i=1
9. 120 10. 130 11. 34 12. 21