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12.2
Arithmetic Sequences and Series
Arithmetic Sequences are lists of numbers that you add (or subtract) the same number to
get each new term (the difference of consecutive terms is constant).
The common difference between each term is denoted as “d”.
The common difference in an Arithmetic Sequence is found by taking one term in the
Arithmetic Sequence and subtracting the previous term from it (or d = an – an – 1)
What is the common difference for each of these Arithmetic Sequences:
1, 4, 7, 10, …
3
14, 10, 6, 2, …
/8, 1/2, 5/8, 3/4, …
The first term of a sequence is denoted as “a1”.
The “nth” term is denoted by “an”.
What is a1 for each of the above sequences?
What is a5 for each of the above sequences?
To find any term of an Arithmetic Sequence use the formula: an = a1 + d (n – 1)
Example 1:
So, for the sequence: 4, 10, 16, 22,… Write the formula to find the nth term, then use it
to find the 20th term.
Example 2:
Now, for an Arithmetic Sequence where the first term is 18, the 12th term is –18, find
the common difference, and then list the first 5 terms. (Use the formula and substitute in
for every value that you know. Think!!)
Example 3:
One term of an Arithmetic Sequence is a19 = 48, the common difference is d = 3. a)
Find the value of the first term; b) write the rule for the nth term; and c) graph the
sequence.
Example 4:
Two terms of an Arithmetic Sequence are a8 = 21 and a27 = 97. Find the common
difference d, the first term a1, and write the rule for the sequence.
An Arithmetic Series, remember is a sum of an Arithmetic Sequence. To find the value
of the sum of an Arithmetic Series, there is a formula so you do not have to add up all
the terms. The formula is:
Sn = n(a1 + an)/2.
What do you think Sn stands for? What does n stand for? What does a1 stand for?
What does an stand for?
Example 1:
Given the Arithmetic Series: 3 + 7 + 11+ 15 +… Find S30.
Example 2:
Find the sum of the Arithmetic Series
20
S (3 + 5i).
i=1