1) The first four terms of the sequence an = n/(1-2n) are -1, -2/3, -3/5, -4/7. 2) The sequence 2/3, -4/9, 8/27, ... is represented by the nth-term rule an = 2/3(-2/3)n-1, which is a geometric sequence. 3) The series Σ(-1)n diverges because it oscillates between -1 and 1 without approaching a specific number.

1. Sequences and Series
A Variety of Problems for You to Practice

2. List the first four terms of the sequence given by
the nth-term rule 𝑎 𝑛 =
𝑛
1−2𝑛
.
•Plug in 1 to get the first term, 2 to get the
second term, and so forth
•The first four terms are −1, −
2
3
, −
3
5
, −
4
7

3. Represent the sequence
2
3
, −
4
9
,
8
27
, … using an nth-
term rule.
• This sequence is geometric, because we multiply by a common
ratio of −
2
3
to get from one term to the next.
• The general form for the nth-term of a geometric sequence is
𝑎 𝑛 = 𝑎1(𝑟) 𝑛−1.
• In this problem, the nth-term rule would be 𝑎 𝑛 =
2
3
−
2
3
𝑛−1
.

4. Determine whether the series 𝑛=1
∞
(−1) 𝑛
converges or diverges. Justify your answer.
• Notice that this is a series, as indicated by the summation notation.
• Notice that this is an infinite series, as indicated by the upper bound
that is infinte.
• If the series converges, it means that the sum of all of its terms
approaches a specific number.
• If the series diverges, it means that the sum of all of its terms does not
approach a specific number.
• Write out a few terms of this series: -1 + 1 – 1 + 1
• Because this series will continue to oscillate back and forth between -1
and 1, this series diverges.

5. Calculate the sum of the infinite series
𝑛=1
∞
3
1
2
𝑛−1
.
• This series is geometric because it is of the form 𝑎1(𝑟) 𝑛−1.
• The common ratio (r) is
1
2
; as long as 𝑟 < 1, there is a sum
• The formula for the sum of a convergent geometric series is
𝑆 =
𝑎1
1−𝑟
.
• For this series, our sum would be:
• 𝑆 =
3
1−
1
2
=
3
1
2
= 6