Learning Activity Sheet Pre-Calculus Quarter 1, Weeks 8: Sequences and Series

LEARNING ACTIVITY SHEET
PRE-CALCULUS
QUARTER 1, WEEKS 8
Name: ________________________________ Date: ________________

Grade & Section: ______________________
SEQUENCES AND SERIES
I. Learning Competency
Differentiate a series from a sequence (STEM_PC11SMI-Ih- 2)
II. Objectives
At the end of the lesson, the student should be able to:
1. recall the different types of sequences and series;
2. differentiate a series from sequence; and
3. solve problems on sequences and series
III. Introduction (Key Concept)

A sequence is a function whose domain is the set of positive integers, while
a series represents the sum of the terms of a sequence.
Many sequences have patterns and through these patterns, you will be able to
identify the different types of sequences.
Arithmetic Sequence
A sequence whose consecutive terms have a common difference is an

arithmetic sequence.
The sequence a1, a2, a2, ….., an is arithmetic, if there is a number d such that
a2 – a1 = d, a3 – a2 = d, a4 – a 3 = d
and so on. The number d is the common difference of the arithmetic sequence.
If the nth term an of an arithmetic sequence with first term a1 and the
common difference d is given by
𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅
then, the associated series with n terms is given by
𝒏 𝒏
𝑺𝒏 = (𝒂𝟏 + 𝒂𝒏 ) or 𝑺𝒏 = {𝟐𝒂𝟏 + (𝒏 − 𝟏)𝒅}
𝟐 𝟐
1
Geometric Sequence
A sequence whose consecutive terms have a common ratio is a geometric

sequence. The sequence a1, a2, a2, ….., an is a geometric, if there is a nonzero number
𝒂𝟐 𝒂𝟑 𝒂𝟒
r such that = r, = r, = r,
𝒂𝟏 𝒂𝟐 𝒂𝟑
The nonzero number r is the common ratio of the geometric sequence.
If the nth term an of a geometric sequence with first term a1 and the common
ratio r is given by 𝒂𝒏 = 𝒂𝟏 ∙ 𝒓𝒏−𝟏
then, the associated series with n terms is given by
𝒂𝟏 − 𝒂𝟏 𝒓𝒏 𝒂𝟏 (𝟏−𝒓𝒏 )
𝑺𝒏 = or 𝑺𝒏 = r≠1
𝟏−𝒓 𝟏−𝒓
𝑺𝒏 = 𝒏𝒂𝟏 . r=1
Moreover, the sum S of an infinite geometric series with -1< r < 1 is given by
𝒂𝟏
𝑺= .
𝟏−𝒓
However, if |𝑟| ≥ 1, then the sum of an infinite geometric series does not
exist.
Fibonacci Sequence
The Fibonacci sequence is named after its discoverer, Leonardo Fibonacci.
1, 1, 2, 3, 5, 8, 13, 21, 34, ……..
In general, the nth term of the Fibonacci sequence is given by
𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒂𝒏−𝟐 where 𝒂𝟏 = 𝒂𝟐 = 𝟏

Harmonic Sequence
A sequence a2, a2, ….., an is a Harmonic if their reciprocals

1 1 1 1
,𝑎 ,𝑎 ,….𝑎 form an arithmetic sequence.
𝑎1 2 3 𝑛
2
IV. Activities
ACTIVITY 1. Name Me
Direction. Determine if the given sequence is arithmetic, geometric, Fibonacci,

harmonic or neither by writing A, G, F, H or N, respectively.
1. 4, 7, 10, 13, 16,…

2. 5, - 5, 5, – 5, 5….
4 4 4 4 4
3. , , , , , ...
5 9 13 16 19
4. 1, 0.1, 0.01, 0.001, …

1
5. 16, 4, 1, , ….
4
6. 4, 8, 20, 56, 16, ….

5 7
7. 2, , 3, , 4, ….
2 2
8. 5, 8, 13, 21, 34, ….

1 2 1 2 1
9. , , , , , ...
2 5 3 7 4
10. 5, −2, −7, −9, ….
ACTIVITY 2. Arithmetic Sequence and Series
Direction. Solve the following problems.
1. Give the next three terms of the sequence 21, 27, 33, 39, …..
2. If a1 = 7 and d = 5, what is the 5th term of the arithmetic sequence?
3. What is the nth term of the arithmetic sequence if a 1 -1, d = -10 and n =
25?
4. Find the sum of the arithmetic series 7, 14, 21, 28, + ……+ 98?
5. What is the common ratio of the geometric sequence 4, 12, 36, ….
6. What are the missing geometric means in the sequence 81, ___,
___, 3.
27
7. If the sequence is 4, 6, 9, ,…, find the equation of the nth term.
2
8. If the first term of a geometric sequence is -3 and the common

1
ratio is 2, what is the sum of the first 5 terms?
3
ACTIVITY 3. Apply and Solve
Direction. Solve each problem.

1 1
1. Give the 3 harmonic means between and .
4 10
2. If the 19th Fibonacci number is 4181 and the 21st Fibonacci number is
10946, What is the 20th Fibonacci number?
3. The auditorium has 21 seats in the first row. Each of the other rows has one
more seat than the row in front of it. If there are 30 rows of seats. What is
the seating capacity of the auditorium?
4. The gardener harvested 16 mangoes on the first day, 48 mangoes in the
second day, and 80 mangoes in the third day. If he continues to harvest
mangoes at this rate, how many mangoes will be harvested in the 10th days?
5
5. Carl Allan drop a rubber ball from a height of 6 ft. It bounces to of its
6
previous height. What is the total distance that the ball travel before it comes
to rest?
ENRICHMENT ACTIVITY.
Knowing Me Knowing You
Geometric Fibonacci
Sequences
Harmonic Arithmetic

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What are the different types of sequences discussed in the document?

What are the different types of sequences discussed in the document?

What is the difference between a sequence and a series?

What is the difference between a sequence and a series?

LEARNING ACTIVITY SHEET

Name: ________________________________ Date: ________________

SEQUENCES AND SERIES

III. Introduction (Key Concept)

A sequence whose consecutive terms have a common difference is an

A sequence whose consecutive terms have a common ratio is a geometric

The nonzero number r is the common ratio of the geometric sequence.

The Fibonacci sequence is named after its discoverer, Leonardo Fibonacci.

1, 1, 2, 3, 5, 8, 13, 21, 34, ……..

In general, the nth term of the Fibonacci sequence is given by

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒂𝒏−𝟐 where 𝒂𝟏 = 𝒂𝟐 = 𝟏

A sequence a2, a2, ….., an is a Harmonic if their reciprocals

Direction. Determine if the given sequence is arithmetic, geometric, Fibonacci,

1. 4, 7, 10, 13, 16,…

4. 1, 0.1, 0.01, 0.001, …

6. 4, 8, 20, 56, 16, ….

8. 5, 8, 13, 21, 34, ….

10. 5, −2, −7, −9, ….

ACTIVITY 2. Arithmetic Sequence and Series

Direction. Solve the following problems.

8. If the first term of a geometric sequence is -3 and the common

Direction. Solve each problem.

Knowing Me Knowing You

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