Introduction
Hi guys, welcome to today’s tutorial. In this lesson, we’ll be treating Sequences and Series, an important topic in mathematics that appears frequently in exams.
Let’s get started with the questions and answers.
Our objective is simple:
- To introduce the concepts of sequence and series.
- To highlight their definitions and applications.
- To explain with examples how they work in problem-solving.
Definition of a Sequence
A sequence is an ordered list of numbers that follow a specific rule or pattern. This means that every number in the sequence is connected by a guiding formula or logic.
Example
Consider the sequence: 1, 3, 5, 7.
- Here, each term increases by 2.
- 1+2=31 + 2 = 31+2=3
- 3+2=53 + 2 = 53+2=5
- 5+2=75 + 2 = 75+2=7
- 7+2=97 + 2 = 97+2=9, and so on.
Thus, the rule guiding this sequence is “add 2 to get the next number.”
Each number in the sequence is called a term.
Series Explained
When the terms of a sequence are added together, it forms a series.
- Example: For the sequence 1, 3, 5, 7, the series becomes:
1+3+5+7=161 + 3 + 5 + 7 = 161+3+5+7=16.
Series are very useful in solving real-life problems such as calculating growth, finance, and progression in patterns.
Arithmetic Progression (AP)
An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant.
For example:
2, 4, 6, 8, 10…
Here, the common difference (d) = 2.
The general formula for the nth term is: Tn=a+(n−1)dT_n = a + (n-1)dTn=a+(n−1)d
Where:
- aaa = first term
- ddd = common difference
- nnn = position of the term
Geometric Progression (GP)
A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant ratio.
For example:
2, 4, 8, 16…
Here, the common ratio (r) = 2.
The nth term is given by: Tn=ar(n−1)T_n = ar^{(n-1)}Tn=ar(n−1)
Exam Tip
Always check if the difference between terms is constant (AP) or if the ratio is constant (GP).
Learn the formulas for both nth term and sum of n terms because these are common in exam questions.
Practice past questions regularly—most exam boards love testing sequences and series!
Watch the full tutorial here: https://www.youtube.com/watch?v=Kmp3Rk8Rbi8
Conclusion
Sequences and Series form the foundation of progression in mathematics. Understanding how to identify patterns, write terms, and calculate sums will give you a strong edge in exams. Whether it’s Arithmetic Progression or Geometric Progression, mastery of this topic is essential for success.
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