1. 11.2 – Arithmetic Sequences

2. How do I know if it is an arithmetic sequence?
Look for a common difference between consecutive terms
Ex: 2, 4, 8,
Common Difference?
2+2 = 4
4+ 4 = 8
8 + 8 = 16
No. The sequence is NOT Arithmetic
Ex: 48, 45, 42,
Common Difference?
= 45
= 42
= 39
Yes. The sequence IS Arithmetic. Subtract 3 each time

3. Important Formulas for Arithmetic Sequence:
Recursive Formula
Explicit Formula
a1=
an = an – 1 + d
an = a1 + (n-1)d
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
d is the common difference
Arithmetic Mean
The average between any arithmetic two values

4. How does this work: Start with the easy one:
Find the missing term of the arithmetic sequence 84, ____ , 110
Strategy: Find the average
= 97 2

5. Ex: Write the explicit and recursive formula for each sequence
2, 4, 6, 8, 10
First term: a1 = 2
difference = 2
an = a1 + (n-1)d
a1 =
an = 2 + (n-1)2
Explicit:
an = an – 1 + 2, where a1 = 2
an = an – 1 + d
Recursive:

6. Explicit Arithmetic Sequence Problem
Find the 25th term in the sequence of , 11, 17, 23,
an = a1 + (n-1)d
Start with the explicit sequence formula
Find the common difference between the values.
difference = 6
a25 = 5 + (25 -1)6
Plug in known values
a25 = 5 + (24)6 =149
Simplify

7. Explicit Arithmetic Sequence Problem
Suppose you have saved $75 towards the purchase of a new iPad. You plan to save at least $12 from mowing your neighbor’s yard each week. In all, what is the minimum amount of money you will have in 26 weeks?
an = a1 + (n-1)d
Start with the explicit sequence formula
Find the common difference between the values. You will save $12 a week so this is your difference.
difference = 12
a26 = 75 + (27 -1)12
Plug in known values
WAIT: Why 27 and not 26 for n ?
The first term in the sequence, 75, came before the weeks started (think of it as week 0). Therefore you want one more week in your formula to account for the $75 that you had before you started saving.
a26 = 75 + (26)12 =$387
Simplify

8. Let’s try one:
Find the 25th term of the arithmetic sequence: 26, 13, 0, -13

9. Explicit Arithmetic Sequence Problem
Find the 25th term of the arithmetic sequence: 26, 13, 0, -13
an = a1 + (n-1)d
Start with the explicit sequence formula
Find the common difference between the values.
difference = -13
a25 = 26 + (25 -1)(-13)
Plug in known values
a25 = 26 + (24)(-13) =-286
Simplify