7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for...
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7.2 Analyze Arithmetic 7.2 Analyze Arithmetic Sequences & Series Sequences & Series p.442 p.442 What is an arithmetic sequence? What is an arithmetic sequence? What is the rule for an arithmetic What is the rule for an arithmetic sequence? sequence? How do you find the rule when given How do you find the rule when given two terms? two terms?
7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for an arithmetic sequence? How do you find the rule when
Text of 7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for...
Slide 1
7.2 Analyze Arithmetic Sequences & Series p.442 What is an
arithmetic sequence? What is the rule for an arithmetic sequence?
How do you find the rule when given two terms?
Slide 2
Arithmetic Sequence: The difference between consecutive terms
is constant (or the same). The constant difference is also known as
the common difference (d). Find the common difference by
subtracting the term on the left from the next term on the
right.
Slide 3
Example: Decide whether each sequence is arithmetic.
-10,-6,-2,0,2,6,10,-10,-6,-2,0,2,6,10, -6--10=4-6--10=4
-2--6=4-2--6=4 0--2=20--2=2 2-0=22-0=2 6-2=46-2=4 10-6=410-6=4 Not
arithmetic (because the differences are not the same)
5,11,17,23,29,5,11,17,23,29, 11-5=611-5=6 17-11=617-11=6
23-17=623-17=6 29-23=629-23=6 Arithmetic (common difference is
6)Arithmetic (common difference is 6)
Slide 4
Rule for an Arithmetic Sequence
Slide 5
Example: Write a rule for the nth term of the sequence
32,47,62,77,. Then, find a 12. There is a common difference where
d=15, therefore the sequence is arithmetic.There is a common
difference where d=15, therefore the sequence is arithmetic. Use a
n =a 1 +(n-1)dUse a n =a 1 +(n-1)d a n =32+(n-1)(15) a n
=32+(n-1)(15) a n =32+15n-15 a n =32+15n-15 a n =17+15n a n =17+15n
a 12 =17+15(12)=197
Slide 6
One term of an arithmetic sequence is a 19 = 48. The common
difference is d = 3. a n = a 1 + (n 1) d a 19 = a 1 + (19 1) d 48 =
a 1 + 18(3) Write general rule. Substitute 19 for n Solve for a 1.
a. So, a rule for the n th term is: a. Write a rule for the nth
term. 6 = a 1 Substitute 48 for a 19 and 3 for d. SOLUTION a. Use
the general rule to find the first term. a n = a 1 + (n 1) d = 6 +
(n 1) 3 Write general rule. Substitute 6 for a 1 and 3 for d.
Simplify.
Slide 7
b. Graph the sequence. One term of an arithmetic sequence is a
19 = 48. The common difference is d =3. Create a table of values
for the sequence. The graph of the first 6 terms of the sequence is
shown. Notice that the points lie on a line. This is true for any
arithmetic sequence. b.
Slide 8
Example: One term of an arithmetic sequence is a 8 =50. The
common difference is 0.25. Write a rule for the nth term. Use a n
=a 1 +(n-1)d to find the 1 st term!Use a n =a 1 +(n-1)d to find the
1 st term! a 8 =a 1 +(8-1)(.25) 50=a 1 +(7)(.25) 50=a 1 +1.75
48.25=a 1 * Now, use a n =a 1 +(n-1)d to find the rule. a n
=48.25+(n-1)(.25) a n =48.25+.25n-.25 a n =48+.25n
Slide 9
Now graph a n =48+.25n. Just like yesterday, remember to graph
the ordered pairs of the form (n,a n )Just like yesterday, remember
to graph the ordered pairs of the form (n,a n ) So, graph the
points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.So, graph the
points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.
Slide 10
Example: Two terms of an arithmetic sequence are a 5 =10 and a
30 =110. Write a rule for the nth term. Begin by writing 2
equations; one for each term given.Begin by writing 2 equations;
one for each term given. a 5 =a 1 +(5-1)d OR 10=a 1 +4d And a 30 =a
1 +(30-1)d OR 110=a 1 +29d Now use the 2 equations to solve for a 1
& d.Now use the 2 equations to solve for a 1 & d. 10=a 1
+4d 10=a 1 +4d 110=a 1 +29d (subtract the equations to cancel a 1 )
-100= -25d So, d=4 and a 1 =-6 (now find the rule) a n =a 1 +(n-1)d
a n =-6+(n-1)(4) OR a n =-10+4n
Slide 11
Example (part 2): using the rule a n =-10+4n, write the value
of n for which a n =-2. -2=-10+4n8=4n2=n
Slide 12
Two terms of an arithmetic sequence are a 8 = 21 and a 27 = 97.
Find a rule for the nth term. SOLUTION STEP 1 Write a system of
equations using a n = a 1 + (n 1)d and substituting 27 for n (Eq 1)
and then 8 for n (Eq 2). STEP 2 Solve the system. 76 = 19d 4 = d 97
= a 1 + 26(4) Subtract. Solve for d. Substitute for d in Eq 1. 27 =
a 1 Solve for a 1. STEP 3 Find a rule for a n. a n = a 1 + (n 1)d
Write general rule. = 7 + (n 1)4 Substitute for a 1 and d. = 11 +
4n Simplify. a 27 = a 1 + (27 1)d 97 = a 1 + 26d a 8 = a 1 + (8 1)d
21 = a 1 + 7d Equation 1 Equation 2
Slide 13
What is an arithmetic sequence?What is an arithmetic sequence?
The difference between consecutive terms is a constant What is the
rule for an arithmetic sequence?What is the rule for an arithmetic
sequence? a n =a 1 +(n-1)d How do you find the rule when given two
terms?How do you find the rule when given two terms? Write two
equations with two unknowns and use linear combination to solve for
the variables.
Slide 14
7.2 Assignment p. 446, 3-35 odd
Slide 15
Analyze Arithmetic Sequences and Series day 2 What is the
formula for find the sum of a finite arithmetic series?
Slide 16
Arithmetic Series The sum of the terms in an arithmetic
sequenceThe sum of the terms in an arithmetic sequence The formula
to find the sum of a finite arithmetic series is:The formula to
find the sum of a finite arithmetic series is: # of terms 1 st Term
Last Term
Slide 17
Example: Consider the arithmetic series 20+18+16+14+. Find the
sum of the 1 st 25 terms.Find the sum of the 1 st 25 terms. First
find the rule for the nth term.First find the rule for the nth
term. a n =22-2na n =22-2n So, a 25 = -28 (last term)So, a 25 = -28
(last term) Find n such that S n =-760Find n such that S n
=-760
Slide 18
-1520=n(20+22-2n) -1520=-2n 2 +42n 2n 2 -42n-1520=0 n 2
-21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive
solution!
Slide 19
SOLUTION a 1 = 3 + 5(1) = 8 a 20 = 3 + 5(20) =103 S 20 = 20 ( )
8 + 103 2 = 1110 Identify first term. Identify last term. Write
rule for S 20, substituting 8 for a 1 and 103 for a 20. Simplify.
ANSWER The correct answer is C.
Slide 20
You are making a house of cards similar to the one shown Write
a rule for the number of cards in the nth row if the top row is row
1. a. House Of Cards SOLUTION Starting with the top row, the
numbers of cards in the rows are 3, 6, 9, 12,.... These numbers
form an arithmetic sequence with a first term of 3 and a common
difference of 3. So, a rule for the sequence is: a. a n = a 1 + (n
1) = d = 3 + (n 1)3 = 3n Write general rule. Substitute 3 for a 1
and 3 for d. Simplify.
Slide 21
You are making a house of cards similar to the one shown What
is the total number of cards if the house of cards has 14 rows? b.
House Of Cards SOLUTION Total number of cards = S 14 Find the sum
of an arithmetic series with first term a 1 = 3 and last term a 14
= 3(14) = 42. b.
Slide 22
5. Find the sum of the arithmetic series (2 + 7i). i = 1
SOLUTION a 1 = 2 + 7(1) = 9 a 12 = 2 + (7)(12) = 2 + 84 = 86 ( ) S
n = n a 1 + a n 2 S 12 = 570 ANSWER S 12 = 570
Slide 23
What is the formula for find the sum of a finite arithmetic
series?