Understanding the relationship between arithmetic sequences and linear equations.

1. The Story of Young Gauss ...
or all about arithmetic series
Photo Source: Karl
Gauss (1777–1855)

2. Write the first five terms of the arithmetic sequence defined by:
g(x) = –3x + 10
f(x) = 2x + 5
Sketch their graphs:

3. Some Definitions
Sequence: An ordered list of numbers that follow a certain pattern (or
rule).
Arithmetic Sequence: (i) Recursive Definition: An ordered list of
numbers generated by continuously adding a value (the
common difference) to a given first term.
(ii) Implicit Definition: An ordered list of numbers where
each number in the list is generated by the equation of a line.
Common Difference (d): (i) The number that is repeatedly added to
successive terms in an arithmetic sequence.
(ii) From the implicit definition, d is the slope of the linear
equation.

4. To Find The Common Difference
d is the common difference d = tn - t(n - 1)
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence
To Find the nth Term In an Arithmetic Sequence
tn is the nth term tn = a + (n - 1)d
a is the first term
n is the quot;rankquot; of the nth term in the sequence
d is the common difference
Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...
Implicitly?
Solution: a = 11 t = 11 + (51 - 1)(-6)
51
d = 5 - 11 t51 = 11 + (50)(-6)
= -6 t51 = 11 - 300
n = 51 t51 = -289

5. Example: Find the 100th term t100 of the sequence 11, 5, -1, -7, ...
tn = a + (n - 1)d
Example: Find the 51st term t 51 of the sequence 11, 5, -1, -7, ...

6. What is the defining linear function that produces the following first
four terms of an arithmetic sequence?
1, 3, 5, 7, . . .
2, 4, 6, 8, . . .

7. Find the number of terms in each of the following arithmetic sequences.
10, 15, 20, ... , 250 40, 38, 36, ... , -30