2. Warm Up
Find the 5th term of each sequence.
1. an = n + 6 2. an = 4 – n
3. an = 3n + 4
Write a possible explicit rule for the nth term
of each sequence.
4. 4, 5, 6, 7, 8,… 5. –3, –1, 1, 3, 5, …
6.
11 –1
19
an = n + 3 an = 2n – 5
3. Find the indicated terms of an
arithmetic sequence.
Find the sums of arithmetic series.
Objectives
5. The cost of mailing a letter in 2005
gives the sequence 0.37, 0.60, 0.83,
1.06, …. This sequence is called an
arithmetic sequence because its
successive terms differ by the same
number d (d ≠ 0), called the
common difference. For the mail
costs, d is 0.23, as shown.
6. Recall that linear functions have a constant first
difference. Notice also that when you graph the
ordered pairs (n, an) of an arithmetic sequence,
the points lie on a straight line. Thus, you can
think of an arithmetic sequence as a linear
function with sequential natural numbers as the
domain.
7. Check It Out! Example 1a
Determine whether the sequence could be
arithmetic. If so, find the common difference
and the next term.
1.9, 1.2, 0.5, –0.2, –0.9, ...
The sequence could be arithmetic with a common
difference of –0.7.
1.9, 1.2, 0.5, –0.2, –0.9
–0.7 –0.7 –0.7 –0.7Differences
The next term would be –0.9 – 0.7 = –1.6.
8. The sequence is not arithmetic because the first
differences are not common.
Check It Out! Example 1b
Determine whether the sequence could be
arithmetic. If so, find the common difference
and the next term.
Differences
9. Each term in an arithmetic sequence is the sum
of the previous term and the common difference.
This gives the recursive rule an = an – 1 + d. You
also can develop an explicit rule for an arithmetic
sequence.
10. Notice the pattern in the
table. Each term is the
sum of the first term and
a multiple of the
common difference.
This pattern can be
generalized into a rule
for all arithmetic
sequences.
11.
12. Check It Out! Example 2a
Find the 11th term of the arithmetic sequence.
–3, –5, –7, –9, …
Step 1 Find the common difference: d = –5 – (–3)= –2.
Step 2 Evaluate by using the formula.
an = a1 + (n – 1)d General rule.
a11= –3 + (11 – 1)(–2)
Substitute –3 for a1, 11 for n,
and –2 for d.
= –23
The 11th term is –23.
13. Check It Out! Example 2a Continued
n 1 2 3 4 5 6 7 8 9 10 11
an –11 –13 –15 –17 –19 –21 –23–9–7–5–3
Check Continue the sequence.
14. Find the 11th term of the arithmetic sequence.
Check It Out! Example 2b
9.2, 9.15, 9.1, 9.05, …
Step 1 Find the common difference:
d = 9.15 – 9.2 = –0.05.
Step 2 Evaluate by using the formula.
an = a1 + (n – 1)d General rule.
a11= 9.2 + (11 – 1)(–0.05)
Substitute 9.2 for a1, 11 for n,
and –0.05 for d.
= 8.7
The 11th term is 8.7.
15. Check It Out! Example 2b Continued
Check Continue the sequence.
n 1 2 3 4 5 6 7 8 9 10 11
an 9.2 9.15 9.1 9.05 9 8.95 8.9 8.85 8.8 8.75 8.7
16. Check It Out! Example 3
Find the missing terms in the arithmetic sequence
2, , , , 0.
an = a1 + (n – 1)d
0 = 2 + (5 – 1)d
–2 = 4d
Step 1 Find the common difference.
General rule.
Substitute 0 for an,
2 for a1, and 5 for n.
Solve for d.
17. = 1
Check It Out! Example 3 Continued
The missing terms are
Step 2 Find the missing terms using d= and a1= 2.
19. Check It Out! Example 4a
Find the 11th term of the arithmetic sequence.
a2 = –133 and a3 = –121
an = a1 + (n – 1)d
a3 = a2 + (3 – 2)d
–121 = –133 + d
d = 12
Step 1 Find the common difference.
Let an = a3 and a1 = a2. Replace 1 with 2.
Simplify.
Substitute –121 for a3 and –133 for a2.
a3 = a2 + d
20. Check It Out! Example 4a Continued
an = a1 + (n – 1)d
–133 = a1 + (2 – 1)(12)
–133 = a1 + 12
–145 = a1
Step 2 Find a1.
General rule
Substitute –133 for an, 2 for n,
and 12 for d.
Simplify.
21. an = a1 + (n – 1)d
a11 = –145 + (n – 1)(12)
= –25
The 11th term is –25.
Step 3 Write a rule for the sequence, and evaluate to
find a11.
General rule.
Substitute –145 for a1 and 12
for d.
Evaluate for n = 11.
Check It Out! Example 4a Continued
a11 = –145 + (11 – 1)(12)
22. Check It Out! Example 4b
Find the 11th term of each arithmetic sequence.
a3 = 20.5 and a8 = 13
an = a1 + (n – 1)d
a8 = a3 + (8 – 3)d
Step 1 Find the common difference.
a8 = a3 + 5d
13 = 20.5 + 5d
–7.5 = 5d
–1.5 = d
General rule
Let an = a8 and a1 = a3.
Replace 1 with 3.
Simplify.
Substitute 13 for a8 and 20.5 for a3.
Simplify.
23. Check It Out! Example 4b Continued
an = a1 + (n – 1)d
20.5 = a1 + (3 – 1)(–1.5)
20.5 = a1 – 3
23.5 = a1
Step 2 Find a1.
General rule
Substitute 20.5 for an, 3 for n,
and –1.5 for d.
Simplify.
24. Check It Out! Example 4b Continued
an = a1 + (n – 1)d
a11 = 23.5 + (n – 1)(–1.5)
Step 3 Write a rule for the sequence, and evaluate to
find a11.
a11 = 8.5
The 11th term is 8.5.
General rule
Substitute 23.5 for a1
and –1.5 for d.
Evaluate for n = 11.a11 = 23.5 + (11 – 1)(–1.5)
25. In Lesson 12-2 you wrote and evaluated series.
An arithmetic series is the indicated sum of the
terms of an arithmetic sequence. You can derive
a general formula for the sum of an arithmetic
series by writing the series in forward and
reverse order and adding the results.
26.
27.
28. These sums are actually partial sums. You cannot
find the complete sum of an infinite arithmetic
series because the term values increase or
decrease indefinitely.
Remember!
29. Check It Out! Example 5a
Find the indicated sum for the arithmetic series.
S16 for 12 + 7 + 2 +(–3)+ …
d = 7 – 12 = –5
a16 = 12 + (16 – 1)(–5)
= –63
Find the common difference.
Find the 16th term.
30. Check It Out! Example 5a Continued
= 16(–25.5)
= –408
Substitute.
Sum formula.
Simplify.
Find S16.
31. Check It Out! Example 5b
Find the indicated sum for the arithmetic series.
a1 = 50 – 20(1) = 30
a15 = 50 – 20(15) = –250
Find 1st and 15th terms.
32. = 15(–110)
= –1650
Check It Out! Example 5b Continued
Find S15.
Substitute.
Sum formula.
Simplify.
33. Check It Out! Example 6a
What if...? The number of seats in the first row
of a theater has 14 seats. Suppose that each
row after the first had 2 additional seats.
How many seats would be in the 14th row?
Write a general rule using a1 = 14 and d = 2.
an = a1 + (n – 1)d
a14 = 11 + (14 – 1)(2)
= 11 + 26
= 37
Explicit rule for nth term
Substitute.
Simplify.
There are 37 seats in the 14th row.
34. Check It Out! Example 6b
How many seats in total are in the first 14 rows?
Find S14 using the formula for finding the sum of the
first n terms.
There are 336 total seats in rows 1 through 14.
Formula for first n terms
Substitute.
Simplify.
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