Section 8.1: Sequences and Series
1. Sequences: Infinite Sequence: {1, 2, 3, . . .}. Finite Sequence: {1, 2, 3, . . . , n}
2. Consider the sequence
a(n) = 2n
or
an = 2n .
List the first five terms.
What is the 10th term?
3. Find the first four terms and the 23rd term of the alternating sequence
an = (−1)n · n2 .
4. Find the first five terms of
an =
n
.
n+1
5. We can consider sequences as points on a plane:
6. Find the general form: (always start at n = 1.)
√ √
(a) 1, 2, 3, 2, . . .
(b) −1, 3, −9, 27, . . .
(c) 2, 4, 8, . . .
1
7. Series: a1 , a2 , . . . , an , . . .
Sum: a1 + a2 + . . . + an + . . .
Partial Sum: Sn = a1 + a2 + . . . + an (nth partial sum).
Sigma Notation:
4
X
(4k + 1) = (4 · 1 + 1) + (4 · 2 + 1) + (4 · 3 + 1) + (4 · 4 + 1)
k=1
8. Example: Consider the series: −2, 4, −6, 8, −10, 12, −14
(a) S1 =
(b) S4 =
(c) S5 =
9. Find the following sums:
P
3
(a)
k=1 5k
(b)
P4
(c)
P11
k k
k=0 (−1) 5
i=8
2+
1
i
10. Write sigma notation for each.
(a) 1 + 2 + 4 + 8 + 16 + 32 + 64
(b) −2 + 4 − 6 + 8 − 10
(c) x +
x2
2
+
x3
3
+
x4
4
2
11. Recursive: a1 = 5, an+1 = 2an − 3 for n ≥ 1 list the first five terms.
12. One of the most famous recursive formulas is given by a description of a rabbit population.
In 1202 Leonardo da Pisa posed and solved the problem: A certain man put a pair of rabbits in
a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair
in a year if it is supposed that every month each pair begets a new pair which from the second month
on becomes productive?
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Section 8.2: Arithmetic Sequences and Series
1. Consider the sequence
2, 5, 8, 11, 14, 17, . . . .
What do you notice about this? Write this sequence recursively.
2. Arithmetic sequence is a sequence if there is a number d (common difference) such that
for any n ≥ 2.
an+1 = an + d
One can also write this recursive definition in the closed form:
an = a1 + (n − 1)d.
3. If the following is an arithmetic sequence identify a1 and d. If not say ‘not arithmetic’:
(a) 4, 9, 14, 19, 24, . . .
(b) 34, 27, 20, 13, 6, −1, −8, . . .
(c) 2, 4, 8, 16, 32, . . .
(d) 2, 25 , 3, 72 , 4, . . .
(e) 4, 9, 14, 19, 23, . . .
1
4. Consider the sequence:
4, 7, 10, 13, . . .
(a) Find the 14th term.
(b) Which term equals 301?
5. Suppose we have an arithmetic sequence with a3 = 8 and a16 = 47. Find a1 and d and the first
four terms of the sequence.
6. Sum of first n terms of an arithmetic sequence:
n(a1 + a2 )
Sn =
2
2
7.
(a) Find S15 for 4, 7, 10, 13, . . .
(b) Find
P130
k=1 (4k
+ 5)
8. Rachel accepts a job starting with hourly wage of $14.25 and a raise of 15 cents per hour every 2
months for 5 years. At the end of 5 years what is her wage?
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Section 8.3: Geometric Sequences and Series
1. Consider the sequence
2, 6, 18, 54, 162, . . . .
What do you notice about this? Write this sequence recursively.
2. Geometric sequence is a sequence if there is a number r (common ratio) such that
for any n ≥ 2.
an+1 = ran
One can also write this recursive definition in the closed form:
an = a1 rn−1 .
3. If the following is an geometric sequence identify a1 and r. If not say ‘not geometric’:
(a) 3, 6, 12, 24, 48, . . .
(b) 1, − 12 , 41 , − 18 , . . .
(c) $5200, $39000, $2925, $2193.75 . . .
(d) 2, 52 , 3, 72 , 4, . . .
1
4.
(a) Consider the sequence: 4, 20, 100, . . .. Find the 7th term.
(b) Consider 64, −32, 16, −8, . . .. Find the 10th term
5. Sum of first n terms of a geometric sequence:
1 − rn
S n = a1 ·
,
1−r
wherer 6= 1.
6.
(a) Find the sum of the first 7 terms 3, 15, 75, 375, . . .
(b) Find
P11
k
k=1 (0.3)
2
7. Consider the infinite series
1
1 1 1
+ + + ... + n.
2 4 8
2
S∞ = 1.
Consider the infinite series
2 + 4 + 8 + . . . + 2n .
S∞ = ∞.
8.
S∞ =
a1
,
1−r
when |r| < 1.
9. Find S∞ if possible:
(a) 1 + 3 + 9 + 27 + . . .
(b) −2 + 1 − 12 + 14 + . . .
3