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ALL WORK (NEATLY ORGANIZED) IN A NOTEBOOK
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Date: _______________________
Period: ______
Series
Recall: A sequence is an ordered list of numbers.
The sum of the terms of a sequence is called a series.
In an arithmetic sequence:
To find the sum of a certain number of terms of a arithmetic sequence:
*given on the regents exam reference sheet
where Sn is the sum of n terms (nth partial sum),
a1 is the first term, an is the nth term.
To find the arithmetic series, the formula for an must be used first (same formula as yesterday)
Model Solution
Ex: Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...

To use the sum formula, an needs to be found first:
a1 = 4, n = 20, d = 2
a20 = 4 + (20 – 1)(2)
a20 = 4 + 19(2)
a20 = 4 + 38
a20 = 42

Now the sum formula can be used.
n = 20, a1 = 4, an = a20 = 42
Exercises:
O. Answer each question
48.) Find the sum of the first 30 terms of 5, 9, 13, 17, ...
49.) Determine the sum of the first 17 terms of the arithmetic sequence whose first 4 terms
are -15, -9, -3, 3
50.) Determine the sum of the first 8 terms of the arithmetic sequence whose first 4 terms
are 8, 11, 14, 17
51.) Find the sum of the arithmetic series 3, 6, 9, .... ,99
52.) Determine the sum of 22, 16, 10, … , -80
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In a geometric sequence:
To find the sum of a certain number of terms of a geometric sequence:
*given on the regents exam reference sheet
where Sn is the sum of n terms (nth partial sum),
a1 is the first term, r is the common ratio.
Model Solution
Ex: Find the sum of the first 8 terms of the sequence -5, 15, -45, 135, ...
a1 = -5, r = -3, n = 8
S8 =
S8 =
S8 =
S8 =
S8 = 8200
Exercises:
P. Answer each question
53.) Determine the sum of the first 15 terms of the geometric sequence 1, 2, 4, 8, ….
54.) Determine the sum of the first 11 terms of the geometric sequence 2, -6, 18, -54, ….
55.) Determine the sum of the first 6 terms of the geometric sequence 1000, 200, 40, 8, ….
56.) Determine the sum of the first 9 terms of the geometric sequence 1, 6, 36, 216, ….
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Recursion:
Recursion is the process of choosing a starting term and repeatedly applying the same process to each term
to arrive at the following term. Recursion requires that you know the value of the term immediately before
the term you are trying to find.
A recursive formula always has two parts:
1. the starting value for a1.
2. the recursion equation for an as a function of an-1 (the term before it.)
Consider the sequence 2, 5, 8, 11, ...
 The recursive formula is

This recursion formula could be used to identify many more terms in the series
Consider the sequence 3, 12, 48, 192, ...
 The recursive formula is

This recursion formula could be used to identify many more terms in the series
The recursion formulas do not always have to be a simple arithmetic or geometric series. As long as it
follows a pattern, it can be worked with
Model Solution:
Ex: Given the recursive formula, write the first 4 terms of the sequence:
a1 = -4
a2 = -4 + 5 = 1
a3 = 1 + 5 = 6
a4 = 6 + 5 = 11
-4, 1, 6, 11
Exercises:
Q. Answer each question
57.) Find the first 4 terms of the sequence
58.) Write the first 5 terms of the sequence
59.) Write a recursive formula for the sequence
9, -18, 36, -72, ...
60.) Write a recursive formula for the sequence
61.) Write a recursive formula for the sequence
5, 11, 23, 47, 95, …