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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?. Arithmetic Sequence:. The difference between consecutive terms is constant (or the same).

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7 2 analyze arithmetic sequences series

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?

arithmetic sequence
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.

example decide whether each sequence is arithmetic
-10,-6,-2,0,2,6,10,…

-6--10=4

-2--6=4

0--2=2

2-0=2

6-2=4

10-6=4

Not arithmetic (because the differences are not the same)

5,11,17,23,29,…

11-5=6

17-11=6

23-17=6

29-23=6

Arithmetic (commondifference is 6)

Example: Decide whether each sequence is arithmetic.
example write a rule for the nth term of the sequence 32 47 62 77 then find a 12
Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12.
  • There is a common difference where d=15, therefore the sequence is arithmetic.
  • Use an=a1+(n-1)d

an=32+(n-1)(15)

an=32+15n-15

an=17+15n

a12=17+15(12)=197

slide6

a.

One term of an arithmetic sequence is a19 = 48. The common difference is d = 3.

a. Write a rule for the nth term.

SOLUTION

a. Use the general rule to find the first term.

an= a1+ (n – 1) d

Write general rule.

a19 =a1+ (19– 1) d

Substitute 19 forn

Substitute 48 for a19 and 3 for d.

48 = a1 + 18(3)

Solve for a1.

– 6 = a1

So, a rule for the nth term is:

Write general rule.

an= a1+ (n – 1) d

= – 6+ (n – 1) 3

Substitute – 6 for a1 and 3 for d.

Simplify.

slide7

b.

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. Graph the sequence. One term of an arithmetic sequence is a19 = 48. The common difference is d =3.

slide8
Example: One term of an arithmetic sequence is a8=50. The common difference is 0.25. Write a rule for the nth term.
  • Use an=a1+(n-1)d to find the 1st term!

a8=a1+(8-1)(.25)

50=a1+(7)(.25)

50=a1+1.75

48.25=a1

* Now, use an=a1+(n-1)d to find the rule.

an=48.25+(n-1)(.25)

an=48.25+.25n-.25

an=48+.25n

now graph a n 48 25n
Now graph an=48+.25n.
  • Just like yesterday, remember to graph the ordered pairs of the form (n,an)
  • So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.
example two terms of an arithmetic sequence are a 5 10 and a 30 110 write a rule for the nth term
Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term.
  • Begin by writing 2 equations; one for each term given.

a5=a1+(5-1)d OR 10=a1+4d

And

a30=a1+(30-1)d OR 110=a1+29d

  • Now use the 2 equations to solve for a1 & d.

10=a1+4d

110=a1+29d (subtract the equations to cancel a1)

-100= -25d

So, d=4 and a1=-6 (now find the rule)

an=a1+(n-1)d

an=-6+(n-1)(4) OR an=-10+4n

slide12

a27 = a1 + (27– 1)d

97 = a1 + 26d

21 = a1 + 7d

a8= a1 + (8– 1)d

STEP 2

Solve the system.

STEP 1

Write a system of equations using an= a1 +(n – 1)dand substituting 27for n(Eq 1) and then 8for n(Eq 2).

STEP 3

Find a rule for an.

Two terms of an arithmetic sequence are a8 = 21 and a27 = 97. Find a rule for the nth term.

SOLUTION

Equation 1

Equation 2

Subtract.

76 = 19d

Solve for d.

4 = d

Substitute for d in Eq 1.

97 = a1 + 26(4)

27 = a1

Solve for a1.

an= a1 + (n – 1)d

Write general rule.

Substitute for a1 and d.

= – 7 + (n – 1)4

= – 11 + 4n

Simplify.

slide13
What is an arithmetic sequence?

The difference between consecutive terms is a constant

  • What is the rule for an arithmetic sequence?

an=a1+(n-1)d

  • 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.

7 2 assignment
7.2 Assignment

p. 446, 3-35 odd

analyze arithmetic sequences and series day 2
Analyze Arithmetic Sequences and Series day 2

What is the formula for find the sum of a finite arithmetic series?

arithmetic series
Arithmetic Series
  • The sum of the terms in an arithmetic sequence
  • The formula to find the sum of a finite arithmetic series is:

Last Term

1st Term

# of terms

example consider the arithmetic series 20 18 16 14
Find the sum of the 1st 25 terms.

First find the rule for the nth term.

an=22-2n

So, a25 = -28 (last term)

Find n such that Sn=-760

Example: Consider the arithmetic series 20+18+16+14+… .
slide18
-1520=n(20+22-2n)

-1520=-2n2+42n

2n2-42n-1520=0

n2-21n-760=0

(n-40)(n+19)=0

n=40 or n=-19

Always choose the positive solution!

slide19

( )

8 + 103

2

S20 = 20

The correct answer is C.

ANSWER

SOLUTION

a1= 3 + 5(1) = 8

Identify first term.

Identify last term.

a20= 3 + 5(20) =103

Write rule for S20, substituting 8 for a1 and 103 for a20.

= 1110

Simplify.

slide20

House Of Cards

a.

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.

Write a rule for the number of cards in the nth row if the top row is row 1.

You are making a house of cards similar to the one shown

SOLUTION

an = a1 + (n– 1) = d

Write general rule.

Substitute 3 for a1 and 3 for d.

= 3 + (n – 1)3

Simplify.

= 3n

slide21

House Of Cards

What is the total number of cards if the house of cards has 14 rows?

b.

You are making a house of cards similar to the one shown

SOLUTION

Find the sum of an arithmetic series with first term a1 = 3 and last term a14 = 3(14) = 42.

b.

Total number of cards = S14

slide22

S12 = 570

ANSWER

( )

a1 + an

Sn = n

2

5. Find the sum of the arithmetic series (2 + 7i).

i = 1

SOLUTION

a1= 2 + 7(1) = 9

a12= 2 + (7)(12) = 2 + 84

= 86

S12 = 570

7 2 assignment1
7.2 Assignment:

p. 446

40-48 all, 63-64