Instructions for “Trout Pond Exploration”
This assignment is worth 14 points and is due
at the end of class on Wednesday, 4/18. The
breakdown of points per problem is as
follows:
Problem #1 – two points; Problem #2 – six
points; Problem #3 – two points; Problem #4 –
four points
Complete sentences are required as answers
for problem #1 and problem #3. For problem
#1, if your conjecture ends up being incorrect,
you will still receive full credit for your answer
as long as it is in a sentence and your
conjecture is reasonable. For problem #4, you
should write an equation that has only “NEXT”
on one side of the equation and the word
“NOW” (along with other things) on the other
side of the equation.
Chapter 11 Warm ups and instructions
11.1 Warm ups and instructions
11.1 Warm up #1
Write the definitions of the following words:
Relation, function, domain, range
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
Write the next five terms of the following
sequences:
1.
𝟐
𝟐
𝟒 𝟔
𝟑 𝟒
𝟖
𝟓
, , , ,⋯
𝟏
𝟐
𝟏
𝟑
𝟏
𝟒
2. – 𝟏, , − , , ⋯
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. 17, 22, 27, 32, 37, …
2.
𝟏
𝟑
𝟐
𝟒
, ,
𝟑
𝟓
,
𝟒
𝟔
,…
3. 𝟒, 𝟕, 𝟏𝟐, 𝟏𝟗, ⋯
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 𝒂𝟏𝟓 .
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
Chapter 11 vocabulary quiz
Write the definition of each of the following
terms: relation, function, domain, range,
sequence, arithmetic sequence, geometric
sequence
Chapter 11 alternate quiz
Page 665 problems 62a, 62b, and 62c
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?
Chapter 11 review warm up #1
Page 695 problems 1 – 6, 9 – 14, 21 – 31, 33
Chapter 11 review warm up #2
Page 696 problems 1 - 9
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