Precalculus
Arithmetic and Geometric Series
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Date________________
Goal: By the end of this lesson, you will be able to find the sum of a finite series.
Recall from yesterday:
To find the nth term of an arithmetic sequence:
tn  t1   n  1 d
To find the nth term of a geometric sequence:
tn  t1 r n1
Definition:
A series is an indicated sum of the terms in a sequence.
Finite Series: 2 + 6 + 10 + 14
Infinite Series:
1 1 1 1
    ...
2 4 8 16
Sum of a Finite Arithmetic Series:
Sum of a Finite Geometric Series:
Sn, nth partial sum of a series: Sum of the first n terms of that sequence.
Example 1: Find the sum of the first 25 terms of the arithmetic series 11 + 14 + 17 + …
Example 2: Find the sum of the first 10 terms of the geometric series 2 – 6 + 18 – 54 + …
Precalculus
Arithmetic and Geometric Series
Example 3: Given the sequence: 14 + 12 + 10 + . . . .
If Sn = -54, find n (the # of terms in the partial sum)
Example 4: Determine the sum of the series: 5 + 35 + 245 + . . . . . . . 4117715
Precalculus
Arithmetic and Geometric Series
Sigma notation
n
Sn   t k 
k 1
Read: “The sum from k = 1 to k = n of tk.”
The k is called the index. (It’s a counter)
Sample Problems:
1) Evaluate the expression by writing the terms and adding them up:
a)
3
 2  5k
k 1
 2  c3
3
b)
k 1
k 1
3
h
c h
c) 2   3k 1
k 1
2) For each problem, write Sn using sigma notation and find the sum of the series.
a) S5 for 9 + 5 + 1+ -3 + . . . . . . . .
b) S100 for
FG IJ FG IJ
H K H K
1
1
1
  
.........
3
6
12
Precalculus
Arithmetic and Geometric Series
Practice
For the following problems:
1) Write in Sigma Notation. 2) Then evaluate each sum using the appropriate Sum formula
1. In a geometric sequence, the first term is 27 and the common ratio is
terms.
2.
3.
5 + 8 + 11 + ……+ 65
1 1 1
1
  ..... 
2 10 50
31250
1
. Find the sum of the first 6
3
Precalculus
Arithmetic and Geometric Series
4. 4 + 5 + 6 + 7 + ….. + 23
4
. Write S10 using sigma notation and evaluate the sum.
9
(Remember: In this problem there are 2 sets of answers because there are 2 possible values for r )
5. For the geometric series t3 = 1, t5 =
6. In an arithmetic sequence, the first term is 10, the common difference is -2, and the sum of the first n
terms is -276. Find the value of n.