Chapter 06 | Sequence
and Series | (First Year Math)
Chapter 06 Sequence and Series covers different types of
sequences and their related means. These concepts are very important for board
exams and MCQs. Understanding these topics helps students solve problems
quickly and accurately.
1. Arithmetic Progression (A.P)
An Arithmetic Progression (A.P) is a sequence in which the
difference between consecutive terms is constant. This constant is called the common
difference (d).
Example:
2, 5, 8, 11, 14 ...
Here,
common difference:
d = 5 - 2 = 3
Formula for nth term:
an = a + (n - 1)d
Formula for sum of n terms:
Sn = n/2 (2a + (n - 1)d)
OR
Sn = n/2 (a + l)
Where:
a= First term
d = common difference
n = number of terms
l = last term
A.P is commonly used in MCQs to find missing terms, sum, or common
difference.
2. Arithmetic Mean (A.M)
The Arithmetic Mean (A.M) is the average of numbers.
Formula:
A.M = (a + b) / 2
If A is the arithmetic mean between a and b:
A = (a + b) / 2
Property:
If
a, A, b are in A.P, then:
A - a = b - A
A.M is often used in MCQs to find missing numbers between two
values.
3. Geometric Progression (G.P)
A Geometric Progression (G.P) is a sequence where each term
is multiplied by a constant value called the common ratio (r).
Example:
2, 6, 18, 54 ...
Here:
r = 6 / 2 = 3
Formula for nth term:
an = a × r^(n - 1)
Formula for sum of n terms:
Sn = a (1 - r^n) / (1 - r) (r ≠ 1)
4. Geometric Mean (G.M)
The Geometric Mean (G.M) between two numbers is the square
root of their product.
Formula:
G.M = √(ab)
If G is the geometric mean between a and b:
G = √(ab)
Property:
If
a, G, b are in G.P, then:
G² = ab
G.M is commonly asked in MCQs and short questions.
5. Harmonic Progression (H.P)
A Harmonic Progression (H.P) is a sequence in which the
reciprocals of the terms form an A.P.
Example:
1, 1/2, 1/3, 1/4 ...
Reciprocals:
1, 2, 3, 4 → A.P
So, original sequence is H.P
There is no direct formula like A.P or G.P, but problems are solved
by converting H.P into A.P using reciprocals.
6. Harmonic Mean (H.M)
The Harmonic Mean (H.M) between two numbers is defined as:
Formula:
H.M = 2ab / (a + b)
Important Relation:
A.M ≥ G.M ≥ H.M
This is a very important formula and frequently asked in exams and
MCQs.
7. Relationship Between A.M, G.M, H.M
For two positive numbers a and b:
A.M ≥ G.M ≥ H.M
OR
(a + b)/2 ≥ √(ab) ≥ 2ab/(a + b)
This relation is very important for conceptual MCQs.
8. Key Differences (Important for MCQs)
A.P:
- Based on addition
- Constant difference
G.P:
- Based on multiplication
- Constant ratio
H.P:
- Based on reciprocals
- Converts into A.P
PDF SOLUTION:
9. Exam Importance
These topics are very important for:
- Board exams
- Entry tests
- MCQs preparation
Most questions are based on:
- Finding nth term
- Finding mean (A.M, G.M, H.M)
- Identifying sequence type
- Applying formulas
10. Quick Revision Formulas
A.P:
an = a + (n - 1)d
Sn = n/2 (2a + (n - 1)d)
A.M:
A = (a + b)/2
G.P:
an = a × r^(n - 1)
Sn = a (1 - r^n)/(1 - r)
G.M:
G = √(ab)
H.M:
H = 2ab/(a + b)
Relation:
A.M ≥ G.M ≥ H.M
Conclusion
Sequence and Series is a very important chapter in First Year
Mathematics. Topics like A.P, A.M, G.P, G.M, H.P, and H.M are frequently
asked in exams and MCQs. By understanding formulas and practicing regularly,
students can easily master this chapter and score high marks.
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