Download (9) Arithmetic Sequences (1).notebook

(9) Arithmetic Sequences (1).notebook
Sequences
Definition: A sequence is a set of numbers in a specific order. 2, 5, 8,…. is an example of a sequence.
Note: A sequence may have either a finite or an infinite number of terms.
The terms of a sequence are the individual numbers in the sequence. If we let a1 represent the first term of a sequence, an represent the nth term, and n represent the term number, then the sequence is represented by a1, a2, a3, ….,an, … 2, 5, 8, 11, 14, ....
In the example above, a1=2, a2=5, a3= 8, etc.
Example: Write the first 3 terms of each sequence.
1. an = n2 + 2
a1 = 12 + 2 a1 = 1 + 2
a1 = 3
Start by plugging in n = 1, then n=
2, etc. Write the terms as a
sequence.
a2 = 22 + 2 a2 = 4 + 2
a2 = 6
a3 = 32 + 2 a3 = 9 + 2
a3 = 11
3, 6, 11
2. an = (n+1)(n+2)
a1 = (1 + 1)(1 + 2) a1 = (2)(3)
a1 = 6
a2 = (2 + 1)(2 + 2) a2 = (3)(4)
a2 = 12
a3 = (3 + 1)(3 + 2) a3 = (4)(5)
a3 = 20
6, 12, 20
Arithmetic Sequences
Definition: An arithmetic sequence is a sequence in which each term, after the first, is the sum of the preceding term and a common difference.
+3 3 3 +3 +3
+ +
Example: In the sequence 2, 5, 8, 11, 14, ...... This is an arithmetic sequence because to get each number in the sequence, we add 3. So, we say the common difference is 3.
REMEMBER: a1 = 2 a2 = 5 a3 = 8 etc. AND d = 3
Practice: Determine if the sequences below are arithmetic sequences. If yes, determine the common difference.
(TIP: figure out if the same number is being added over and over again)
1.) ­2, 3, 8, …..
This is an arithmetic
sequence because
5 is added to the first
term to determine the second term,
5 is added to the second term to determine the third term, etc
2.) 5, 9, 13, …..
arithmetic sequence
3.) 1, 7, 14, ……
not an
arithmetic sequence
d = 4
6 is added to find
the second term and
then 7 is added to find the third term
d = 5
4.) 5, ­1, ­7, …….
arithmetic sequence
d = ­6
5.) 3, 5, 8, ......
6.) ­9x, ­2x, 5x, …….
not an
arithmetic sequence
arithmetic sequence
d = 7x
(9) Arithmetic Sequences (1).notebook
How to write an arithmetic sequence:
Practice: Write the first five terms of the arithmetic sequence in which a1 and d are given as follows
1.
= 19, d = 6
Start with a1 and add the common difference (d) to the term. Continue adding to find more terms
a1 = 19
a2 = 19 + 6 = 25
a3 = 25 + 6 = 31
a4 = 31 + 6 = 37
a5 = 37 + 6 = 43
2.
3.
= 27, d = ­4
a1 = 27
a2 = 27 ­ 4 = 23
a3 = 23 ­ 4 = 19
a4 = 19 ­ 4 = 15
a5 = 15 ­ 4 = 11
= 3, d = a1 = 3
3
__
a2 = 3 + =
2
9
__
2
9
3
__
__
a3 = + = 6
2
2
3
__
a4 = 6 + =
2
15
__
2
3
15
__
__
a5 = + = 9
2
2
How to find any term of an arithmetic sequence
If a1 is the first term of an arithmetic sequence, an the nth term, d is the common difference, a formula for finding the value of the nth term of an arithmetic sequence is:
an = a1 + (n ­ 1)d
Steps in Solution
1. List the values of those variables in the formula which are known, and indicate the variable whose value is to be determined.
2. Substitute the known values in the formula
for an, and compute the value to be determined
1. Find the 75th term of the sequence 2, 5, 8,……
Find d first !!!
a1 = 2
d = 3
n = 75
an = a1 + (n ­ 1)d
a75 = 2 + (75 ­ 1)(3)
a75 = 2 + (74)(3)
a75 = 2 + 222
a75 = 224
2. Find the 13th term of 2, 8, 14, 20, 26, …..
a1 = 2
d = 6
n = 13
an = a1 + (n ­ 1)d
a13 = 2 + (13 ­ 1)(6)
a13 = 2 + (12)(6)
a13 = 2 + 72
a13 = 74
3. Find the 43rd term of ­19, ­15, ­11, …..
a1 = ­19
d = 4
n = 43
an = a1 + (n ­ 1)d
a43 = ­19 + (43 ­ 1)(4)
a43 = ­19 + (42)(4)
a43 = ­19 + 168
a43 = 149
(9) Arithmetic Sequences (1).notebook
4. Find the common difference (d) in the arithmetic sequence whose 1st term is 4 and whose 11th term is 64.
a1 = 4
a11 = 64
n = 11
an = a1 + (n ­ 1)d
a11 = a1 + (n ­ 1)d
64 = 4 + (11 ­ 1)(d)
64 = 4 + 10d
60 = 10d
6 = d
5. Write the arithmetic sequence whose first term is 5 and whose
7th term is 17.
Find d using the formula!!!
a1 = 5
a7 = 17
n = 7
an = a1 + (n ­ 1)d
a7 = a1 + (n ­ 1)d
17 = 5 + (7 ­ 1)(d)
17 = 5 + 6d
12 = 6d
2 = d
Then write the sequence!
5, 7, 9, 11, 13, 15, 17, ....