Introduction
-
What have you observe in the images below?
Have you noticed an order? Have you noticed patterns? The patterns you observed is called sequence. A sequence is a succession of numbers in a specific order. Each number in a sequence is called a term. The terms are formed according to some fixed rule or property. They are arranged as the first term, the second term, the third term, and so on. There are different types of sequence, it includes the finite sequence and infinite sequence.
Finite sequence is a sequence with a definite number of terms. It is a sequence that have first and last term identified.
The following sequence are finite:
| Sequence | First Term | Last Term | Number of Terms |
|---|---|---|---|
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 1 | 10 | 10 |
| -3, -9, -27, -81, -243, -729 | -3 | 729 | 6 |
| -1, 1, -1, 1, -1, 1, -1, 1 | -1 | 1 | 8 |
| 100, 90, 80, 70, 60 | 100 | 60 | 5 |
| 267, 262, 257, 252, 247, 242, 237, 232, 227, 222, 217, 212 | 267 | 212 | 12 |
The first and last terms of a sequence are referred to as extremes. The terms between the first and the last terms are called means.
Infinite sequence is a sequence with no definite number of terms.
The following are examples of an infinite sequence.
| Sequence |
|---|
| -9, -2, 5, 12, 19, ... |
| 27, 9, 3, 1, 1/3, 1/9, ... |
| 1/2, 1/4, 1/6, 1/8, ... |
| ..., -64, -49, -36, -25, -16 |
| 1, 4, 9, 16, 25, ... |
A sequence is a function whose domain is the set of natural numbers or a subset of consecutive positive numbers.
Example 1.
Use the functional relation F(n) = 3n+2, where n is a natural number, to write an infinite sequence.
Solution:
If n=1, F(1) = 3(1)+2 = 3+2 = 5 If n=4, F(4) = 3(4)+2 = 12+2 = 14
If n=2, F(2) = 3(2)+2 = 6+2 = 8 If n=5, F(5) = 3(5)+2 = 15+2 = 17
If n=3, F(3) = 3(3)+2 = 9+2 = 11 If n=6, F(6) = 3(6)+2 = 18+2 = 20
The sequence is 5, 8, 11, 14, 17, 20, ...
Example 2.
Using the consecutive positive integers n= 5,6,7,8,9 write the first five terms of the sequence defined by 
Solution:
If n=5, 
If n=8, 
If n=6, 
If n=9, 
If n=7, 
The sequence is 
, 
, 
, 
, 
.
A series is the indicated sum of the terms of a sequence. A series can be denoted by 
, where n refers to the number of terms. The simplified sum of a series for a specific value of n is called the value of the series.
Example 3.
Find the indicated value for each series.
a. 2+4+6+8+...,
b. 
Solution:
First, look for the pattern to identify the other terms until the nth term. Then express these terms as a sum.
a. = 2+4+6+8+10+12+14+16+18+20 = 110
b. 
= 
Arithmetic Sequence and Arithmetic Series
A sequence where the difference between any two consecutive terms is constant. This constant is called the common difference and the said sequence is called arithmetic sequence. An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant term called the common difference. Moreover, the expression denoting the sum of the terms of an arithmetic sequence is called arithmetic series. An arithmetic series is finite if its corresponding sequence is finite. Otherwise, it is infinite.
Arithmetic Sequence
A sequence with a common difference is an arithmetic sequence. The nth term of a sequence is
where 
is the first term, n is the number of terms, and d is the common difference. The terms of an arithmetic sequence that are between two given terms are called arithmetic means.
Arithmetic Series
The sum of the first n terms of an arithmetic series is
where is the first term, 
is the nth term, n is the number of terms, and d is the common difference.
Watch the video below for the better understanding about arithmetic sequence and arithmetic series.
Geometric Sequence and Geometric Sequence
If the arithmetic sequences are formed by addition, geometric sequences are formed by multiplication. Each term in a geometric sequence is found by multiplying the previous term by the same number. The constant multiplier in a geometric sequence is called the common ratio. Moreover, the indicated sum of a geometric sequence is called geometric series.
A geometric sequence with a definite number of terms is referred to as a finite geometric sequence. On the other hand, a geometric sequence with indefinite number of terms is described as an infinite geometric sequence.
Geometric Sequence
A sequence with a common ratio is a geometric sequence. The nth term of the sequence is
where is the first term, n is the number of terms, and r is the common ratio. The terms of an geometric sequence that are between two given terms are called geometric means.
Finite Geometric Series
The sum of the first n terms of a geometric sequence is
where is the first term, is the nth term, n is the number of terms, and r is the common ratio, and r ≠ 1.
Infinite Geometric Series
The sum S of an infinite geometric sequence is
where is the first term and r is the common ratio, and |r|<1.
Watch the video below to know more about geometric sequence and geometric series.
Task
For your individual task, you are expected to:
- Answer the exercises below.
- Create an illustration that shows the relation and connection of the arithmetic sequences and series into real-life situation.
- Write an essay, poem or short story that refers to the similarities and differences of arithmetic sequence and geometric sequence.
Process
Part 1.
A. Write the first five terms of the sequence defined by the following functions.
- F(n) = 3n+2n
- F(n) =
- F(n) =
B. Name the next 5 terms for each arithmetic sequence, given the first term and the common difference.
C. Write the first n terms of the geometric sequence described in each of the following:
D. Find the specified partial sum for each arithmetic series.
- 3+6+9+12+...;
E. Find the indicated sum for the following geometric series.
- 3+12+48+...;
- 1600+ 800+400+200+...;
Part 2.
Create an illustration about a real-life situation where arithmetic sequence and series can be seen. Show through a drawing or symbol or any visual representation the relation and connection of our topic to the reality. You can use any art materials.
Part 3.
Express the similarities and differences of an arithmetic sequences/series and geometric sequences/series with the use of essay, poem or short story. (Pick only one among the three choices).
Format: Times New Roman, 12, All sides 1" margin.
Email your digital copy to princessjenielle21@gmail.com (File name format: SURNAME, First Name - Course/Yr)
Evaluation
Part 2 and Part 3 will be evaluated using this:
| Criteria | Points |
|---|---|
| Creativity | 25 |
| Originality | 20 |
| Relevance | 25 |
| Aligned to the instruction | 30 |
| Total | 100 |
Conclusion
Credits
Reference:
Mathematics Grade 10 Module
Teacher Page
This WebQuest was created by Princess Jenielle B. Forte (BSED4-Mathematics) for the Learning Task 3 in Technology in Teaching and Learning 2.