Download Math 121. Sequences and Summation Notation (Section 11.1) Fall

Math 121.
Sequences and Summation Notation (Section 11.1)
Names.
1. Find the first three terms and 6th term of a sequence whose nth term is given by
an =
(−1)n−1
1 − 9n
Solution: The first three terms are
a1 =
(−1)1−1
1
1
=
=−
1 − 9(1)
−8
8
a2 =
(−1)2−1
−1
1
=
=
1 − 9(2)
−17
17
a3 =
(−1)3−1
1
1
=
=−
1 − 9(3)
−26
26
and the 6th term is
a6 =
−1
1
(−1)6−1
=
=
1 − 9(6)
−53
53
2. Find the 34th term of the sequence whose nth term is defined by
an =
(n + 2)!
, n≥1
(n − 1)!
Solution: The 34th term is (notice there is a lot of cancellation!)
a34 =
36!
36 · 35 · 34 · 33 · 32 · · · 3 · 2 · 1
(34 + 2)!
=
=
= 36 · 35 · 34 = 42840
(34 − 1)!
33!
33 · 32 · · · 3 · 2 · 1
3. Evaluate the following sum
7
X
(−1)n (5n)
n=4
Solution: The expanded sum is
(−1)4 (5)(4) + (−1)5 (5)(5) + (−1)6 (5)(6) + (−1)7 (5)(7)
which simplifies to −10.
Fall 2016
4. (a) Write an expression for an , the nth term of the sequence of even numbers
(b) Write an expression for bn , the nth term of the sequence of odd numbers
2, 4, 6, 8, 10, 12 . . .
1, 3, 5, 7, 9, 11, . . .
(c) Write an expression for cn , the nth term of the sequence whose first six terms are
5, 7, 9, 11, 13, 15, . . .
(d) Write an expression for dn , the nth term of the sequence whose first six terms are
6, − 8, 10, − 12, 14, − 16, . . .
(e) Write an expression for en , the nth term of the sequence whose terms are multiples of 7, so its first six
terms are
7, 14, 21, 28, 35, 42, . . .
(f) Write an expression for fn , the nth term of the sequence whose first six terms are
4, 11, 18, 25, 32, 39, . . .
Solution: (a) an = 2n
(b) bn = 2n − 1
(c) Each cn results from adding 3 to each even number, so cn = 2n + 3.
(d) The absolute value of each dn is obtained by adding 4 to each even number, and the signs of dn are
negative on the even terms, so dn = (−1)n+1 (2n + 4).
(e) en = 7n
(f) The terms fn are obtained by subtracting 3 from each en and so fn = 7n − 3.
5. Express the following sum in summation notation.
5
6
7
8
+
+
+
9 16 25 36
Solution: Observe the numerator is n + 4 while the denominator is (n + 2)2 for n = 1, . . . , 4. Therefore,
4
X
6
7
8
n+4
5
+
+
+
=
9 16 25 36 n=1 (n + 2)2
There are other ways to express this sum.
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