7 .2 Analyze Arithmetic Sequences & Series

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7 .2 Analyze Arithmetic Sequences &amp; 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

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).
• 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.

-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 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

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.

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.

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 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 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

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.

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

p. 446, 3-35 odd

Analyze Arithmetic Sequences and Series day 2

What is the formula for find the sum of a finite 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

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+… .
-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!

( )

8 + 103

2

S20 = 20

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.

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

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

S12 = 570

( )

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 Assignment:

p. 446

40-48 all, 63-64