1. Lesson 10.2: Arithmetic Sequences & Series
Algebraic formula specific terms
Recursion or Recursive formulas
Arithmetic means
Series
Sum of finite arithmetic sequences

2. Arithmetic Sequence: a sequence whose consecutive terms
have a common difference, d.
d = 4
1. 5, 9, 13, 17, _____
d =
2. 2, -5, -12, -19, _____
The formula for an can be written given a1 and d:
an = a1 + (n – 1)d
= a1 + dn – d
= dn + (a1 – d)
simplified form
an = dn + c (where c = a1 – d)
For 1: a1 = 5, d = 4, so
an = 5 + (n – 1)4 =
For 2: a1 = 2, d = -7,
so an =

3. Write the explicit formula for an
an = a1 + (n – 1)d
an = dn + c (where c = a1 – d)
Write the explicit formula for an
3. a3 = 63, d = 4
a3 = dn + c
so, an = 4n + 51
63 = 4*3 + c
51 = c
p675 #30. a5 = 190, a10 = 115
a10 = a5 + (10 – 5)d
a5 = dn + c
190 = -15*5 + c
265 = c
115 = d
-75 = 5d
-15 = d
find d
so, an = -15n + 265

4. Write the first 5 terms of the arithmetic sequence.
(need the formula or a1 & d)
an = a1 + (n – 1)d
an = dn + c (where c = a1 – d)
1. an = 4n + 51
Have formula, so use it!
a1= 4(1) + 51 = 55
a2 = 4(2) + 51 = 59
etc.
Sequence:
p675 #39. a8 = 26, a12 = 42
No formula, so find a1 & d
a1 = -2
a2 = a1 + d = = 2
a3 = a2 + d =
a4 = a3 + d =
a5 = a4 + d =
a12 = a8 + (12 – 8)d
a8 = a1 + (n – 1)d
find d
find a1
Sequence:

5. A recursion formula defines an in terms of previous term(s).
For arithmetic sequences, an = an-1 + d
(must also provide a term in the sequence).
d = 4
1. 5, 9, 13, 17, _____ 21
a1 = 5, an = an-1 + 4
2. 2, -5, -12, -19, _____
- 26
d = -7
3. a3 = 63, d = 4
p674 #30. a5 = 190, a10 = 115
d = -15

6. 1. Find the 4 arithmetic means between 72 and 52.
Arithmetic means are terms between any nonconsecutive terms in an arithmetic sequence.
an = a1 + (n – 1)d
an = dn + c (where c = a1 – d)
Need d.
1. Find the 4 arithmetic means between 72 and 52.
72, , , , , 52 a1
a6 = a1 + (n – 1)d a6
52 = 72 + (5)d
-20 = 5d
-4 = d
So, 72, 68, 64, 60, 56, 52
2. Find the 3 arithmetic means between 21 and 45.
d = 6, so 21, 27, 33, 39, 45

7. Arithmetic series: the sum of a finite arithmetic sequence.
The sum of a finite arithmetic sequence is or
See derivation on p670. note: an = a1 + (n – 1)d
Examples:
1. Find the sum of the 1st 20 even integers beginning at 8.
Have n = , a1 = , d = , so use 2nd formula.
2. Find
Formula looks like an arithmetic series,
expand a few terms to be sure:
2*3 + 1 = 7, 2*4 + 1 = 9, 2*5 + 1 = 11, …
be careful!
Can easily determine n, a1, & an, so use 1st formula.
n =
a1 = [k = 3]
a108 = [k = 110]

8. Homework:
Arithmetic Sequences:
pp # 2, 5, 7, 20, 24, 34, 38, 62, 64
Arithmetic Series: p675 #42, 44, 48, 50, 54, 66, 70, 72