Chapter 11 Warm ups and instructions
11.1 Warm ups and instructions
11.1 Warm up #1
Write the next five terms of the following
sequences:
1.
𝟐
𝟐
𝟒 𝟔
, ,
𝟑 𝟒
2.
𝟏
– 𝟏,
𝟐
𝟖
,
𝟓
𝟏
,−
𝟑
,⋯
𝟏
,
𝟒
,⋯
Definitions
Sequence – a function whose domain is a set
of consecutive integers
Finite sequence – a sequence that has a last
term
Infinite sequence – a sequence that continues
without stopping
Series – the expression that results when the
terms of a sequence are added
11.1 Warm up #2
Write the next term of the sequence. Then
write a rule for the nth term. Start with an
input of 1.
1. 4, 7, 12, 19, …
2.
𝟏
𝟐
,
𝟑
𝟓
3.
𝟏
𝟐
,
−𝟏
𝟒
,
𝟕
𝟏𝟏
,…
, ,
−𝟏
𝟏𝟔
,⋯
,
𝟓
𝟖
𝟏
𝟖
Summation notation – see page 653
11.1 Warm up #3
Write the series using summation notation.
1. 𝟕 + 𝟏𝟎 + 𝟏𝟑 + 𝟏𝟔 + 𝟏𝟗
2. (−𝟐) + 𝟒 + (−𝟖) + 𝟏𝟔 + (−𝟑𝟐) + ⋯
Find the sum of the series.
3. ∑𝟕𝒊=𝟏 𝟑𝒊 − 𝟒
4. ∑𝟔𝒌=𝟏
𝒌
𝒌+𝟏
5. ∑𝟓𝒏=𝟏(𝒏𝟐 − 𝟏)
11.2 Warm ups and instructions
Definition
Arithmetic sequence – a sequence with a
common difference.
nth term rule for an arithmetic sequence with
first term 𝒂𝟏 and common difference d
𝐚𝐧 = 𝐚𝟏 + (𝐧 − 𝟏)𝐝
11.2 Warm up #1
Write a rule for the nth term of the arithmetic
sequence. Simplify the rule. Then find 𝒂𝟏𝟕 .
1. 36, 32, 28, 24, 20, …
2.
𝟒
𝟓
𝟔
, 𝟏,
𝟓
𝟕
,
𝟓
𝟖
,
𝟓
,⋯
3. 𝒅 = 𝟗, 𝒂𝟔 = 𝟔𝟔
4. 𝒂𝟒 = 𝟏𝟔, 𝒂𝟏𝟎 = 𝟒𝟔
5. 𝒂𝟔 = −𝟑𝟖, 𝒂𝟏𝟏 = −𝟕𝟑
Sum of a finite arithmetic series
𝐚𝟏 + 𝐚𝐧
𝐒𝐧 = 𝐧 (
)
𝟐
11.2 Warm up #2
Find the sum.
1. 𝟒 + 𝟕 + 𝟏𝟎 + 𝟏𝟑 + ⋯ for n = 10
2. ∑𝟏𝟖
𝒊=𝟏(𝟏𝟒 − 𝟔𝒊)
Find the value of n
3. ∑𝒏𝒊=𝟏(−𝟓 + 𝟕𝒊) = 𝟒𝟖𝟔
11.3 Warm ups and instructions
Definition
Geometric sequence – a sequence with a
common ratio
Rule for a geometric sequence
The nth term of a geometric sequence with
the first term 𝒂𝟏 and common ratio 𝒓 is given
by: 𝒂𝒏 = 𝒂𝟏 𝒓𝒏−𝟏
11.3 Warm up
1 – 2 Write a rule for the nth term of the
sequence. Then find 𝒂𝟗 .
1. 𝟑, 𝟏𝟐, 𝟒𝟖, 𝟏𝟗𝟐, ⋯
2.
−𝟒
𝟑𝟔, −𝟏𝟐, 𝟒, , ⋯
𝟑
3 – 4 Write a rule for the nth term of the
geometric sequence.
3. 𝒂𝟒 = −𝟏𝟐, 𝒓 =
4. 𝒂𝟑 =
𝟕
, 𝒂𝟓
𝟒
=
−𝟏
𝟒
𝟕
𝟏𝟔
The sum of a finite geometric series
The sum of the first n terms of a geometric
series with common ratio 𝒓 ≠ 𝟏 is:
𝟏 − 𝒓𝒏
𝑺𝒏 = 𝒂𝟏 (
)
𝟏−𝒓
Find the sum of the geometric series.
5.
𝟑 𝒊−𝟏
𝟏𝟐
∑𝒊=𝟏 𝟖 ( )
𝟐
6.
𝟏 𝐣
𝟕
∑𝐣=𝟎 𝟏𝟐 (− )
𝟐
11.4 Warm ups and instructions
11.4 Warm up #1
Find 𝑺𝟓 , 𝑺𝟏𝟎 , 𝑺𝟓𝟎 , and 𝑺𝟏𝟎𝟎 for the following
infinite series.
𝟑 𝟑 𝟑 𝟑
𝟑+ + + +
+⋯
𝟐 𝟒 𝟖 𝟏𝟔
The sum of an infinite geometric series with
first term 𝒂𝟏 and common ratio 𝒓 is given by
𝒂𝟏
𝑺=
𝟏−𝒓
Provided that |𝒓| < 1. If |𝒓| ≥ 𝟏, the series
has no sum.
11.4 Warm up #2
Find the sum, if it exists
∞
−𝟏 𝐢−𝟏
∑𝟐( )
𝟒
𝐢=𝟏
∞
𝟏 −𝟏𝟎 𝐢
∑ (
)
𝟐 𝟗
𝐢=𝟎
A person is given one push on a tire swing
and then allowed to swing freely. On the
first swing, the person travels a distance of
14 feet. On each successive swing, the
person travels 80% of the distance of the
previous swing. What is the total distance
the person swings?
Review bullet points for the Chapter 11 test
 Definitions: sequence, finite sequence,
infinite sequence, series, arithmetic
sequence, geometric sequence
 Determining whether a sequence is
arithmetic, geometric, or neither and
giving a reason for your answer
 Finding rules for the nth term of a
sequence
 Finding sums of finite and infinite series
 Word problems involving sequences and
series
Chapter 11 review warm up
Problem 16 on page 697 (do parts a – e)
Problem 17 on page 697 (do parts a, b, c, and
f)