Download Sequences and Series I. What do you do when you see sigma

Sequences and Series
I. What do you do when you see sigma notation (Σ)?
1. Σ tells you to take the sum of the terms starting with the number
below the sigma and up through the number above the sigma.
Examples.
If there is a number in front of the sigma notation, you multiply the
entire sum by that constant.
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II. Arithmetic Sequences and Series
a. Sequences - An arithmetic sequence has a common difference,
d, between each successive term.
6, 10, 14, 18, …
d = +4
-2, -5, -8, -11, …
d = -3
To find an individual term in an arithmetic sequence you may use
the formula
An = A1 + (n -1)d
Where An is the nth term, A1 is the first term and d is the common
difference.
Examples.
1. Find the 213th term in the arithmetic sequence where A1 = 9
and
d = 3/2.
2. The 17th term is an arithmetic sequence is 45 and the 25rd term
in the arithmetic sequence is 118.6. Find d and the first term.
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b. Arithmetic Mean - is the average of two numbers
What do you do if you want to find the arithmetic means between
numbers?
Find the common difference.
Examples.
1. Find 3 arithmetic means between 4 and 70.
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Find 4 arithmetic means between -14 and -89.
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c. Arithmetic Series
An arithmetic series is the sum of all indicated terms in a given
sequence, defined as Sn (the partial sum). The formula for the
partial sum of an arithmetic sequence is given on the formula
sheet.
Sn = (n/2)(A1 + An)
Examples.
1. Find the sum of the first 72 terms of the sequence where
the 5th term is 62 and the 10th term is 30.
2. Given an arithmetic sequence with A14 = 12 and A96 =278.5, find the value of A1, d, A123 and S200. 4
III. Geometric Sequences and Series
a. A geometric sequence has a common ratio, r, between each
successive term. To find r, divide a term by the previous term.
r = 4/3
3, 4, 16/3, 64/9, …
Example:
To find the nth term in a geometric sequence you may use the formula
An = A1rn - 1
Where An is the nth term, A1 is the first term, r is the common ratio, and
n is the position of the term.
1. Find the 9th term of the geometric sequence if the
first term is 8 and r = - ½ .
2. The first term of a geometric sequence is 2 and
the 5th term is 162, what is the 8th term?
3. In a geometric series A1 = 5 and A10 = 40, using
the formula for finding the nth term in a geometric
sequence find A37.
4. In a geometric sequence A1 = 10,000 and
r = 2/5, what term is 8192/3125? Only an algebraic
solution will be accepted for full credit.
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B. Geometric Means - The geometric mean is also called the mean
proportional. The geometric mean of two positive numbers is the
principal square root of their product.
Examples:
1. Find three geometric means between 3 and 12 and state the value of
r.
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C. Geometric Series
A geometric series is the sum of the terms of an indicated geometric
sequence.
To find the sum you may use the formula (given on your formula
sheet)
where A1 is the first term, r is the common ratio, and n is the
position of the term.
Examples.
1. In a geometric sequence, A7 = 42 and r = 2, find the exact
values for A1 and the sum of the first 7 terms.
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2. Find the sum of the first eight terms of the geometric series if
the first term is 2 and fifth term is 1250.
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