Satyawati secondary school
Vyas-2, Damauli
Project work: 2083
Submitted By: Susan
Shrestha
Submitted To: Eak Bahadur
Sir
Roll No: 10 ‘Ten’
Subject: Mathematics
Department: Engineering
Teacher’s sign
Qn 1. Geometric sequence and series with some applications
A geometric sequence is a sequence of numbers where each term is obtained by
multiplying the previous term by the same constant number. This constant is called the
common ratio.
For example:
3, 6, 12, 24, 48, …
Here,
𝑟=
6 12
=
=2
3
6
So, the common ratio is 2.
General Term
If the first term is 𝑎and the common ratio is 𝑟, then:
𝑇𝑛 = 𝑎𝑟 𝑛−1
For example, for 2,4,8,16, …,
𝑇𝑛 = 2(2)𝑛−1 = 2𝑛
The sum of the first 𝑛terms is:
𝑆𝑛 =
𝑎(𝑟 𝑛 − 1)
, 𝑟≠1
𝑟−1
Real-Life Applications
Geometric sequences and series are useful whenever something increases or
decreases by a fixed ratio or percentage.

Banking: Compound interest increases money by a fixed percentage.

Depreciation: The value of a vehicle or machine decreases by a fixed
percentage each year.

Population: Population growth can be modeled using a constant growth rate.

Bouncing ball: Each bounce may reach a fixed fraction of the previous height.

Business: Repeated percentage growth in sales or investment can be
represented using geometric sequences.
Qn 2. Arithmetic sequence and series with some applications
An arithmetic sequence represents a pattern in which a fixed amount is
added or subtracted from one term to get the next term.
For example: 10, 15, 20, 25, 30,…
Therefore, the 𝑛𝑡ℎ term is:
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
Its sum is:
𝑆𝑛 =
𝑛
[2𝑎 + (𝑛 − 1)𝑑]
2
If the last term is 𝑙, then:
𝑆𝑛 =
𝑛
(𝑎 + 𝑙)
2
Applications in Daily Life

Monthly savings: Saving Rs. 1,000 more each month than the
previous month.

Salary: Receiving a fixed yearly increment in salary.

Stadium seating: Each row having a fixed number of extra seats.

Staircase design: Each level increasing by a fixed height.

Loan payments: Regular payments following a fixed increase or
decrease.

Production: A factory increasing its daily production by a fixed
number of units.
Qn 3. Various types of functions and their real life applications
A function is a mathematical relationship in which every input has exactly one
corresponding output. Functions are used to represent relationships between different
quantities in everyday life.
1. Constant Function: A function whose value remains the same for every input.
𝑓(𝑥) = 5
Application: A fixed monthly subscription fee that does not change with usage.
2. Identity Function: A function in which the output is equal to the input.
𝑓(𝑥) = 𝑥
Application: Converting a quantity without changing its value, such as 1kg remaining
1kg.
3. Linear Function: A function whose graph is a straight line.
𝑓(𝑥) = 𝑚𝑥 + 𝑐
Application: Calculating the total cost of a taxi when there is a fixed charge plus a
constant charge per kilometre.
4. Quadratic Function: A function involving the square of the variable.
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Application: Finding the height or path of a ball thrown into the air.
5. Polynomial Function: A function made up of powers of a variable with constants.
𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + ⋯ + 𝑎1 𝑥 + 𝑎0
Application: Used in engineering, physics, and modelling changing quantities.
6. Exponential Function: A function in which the variable occurs in the exponent.
𝑓(𝑥) = 𝑎𝑏 𝑥
Application: Compound interest, population growth, and spread of information.
7. Rational Function: A function expressed as the ratio of two polynomial functions.
𝑓(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
Application: Used to calculate quantities such as speed
Qn 4. Finding general terms of various sequences
The general term 𝑇𝑛 gives the value of any term of a sequence by using its position 𝑛.
1. Arithmetic Sequence
Let the first term be 𝑎and common difference be 𝑑.
The sequence is:
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
So,
𝑇1 = 𝑎
𝑇2 = 𝑎 + 𝑑
𝑇3 = 𝑎 + 2𝑑
𝑇4 = 𝑎 + 3𝑑
Therefore,
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
2. Geometric Sequence
Let the first term be 𝑎and common ratio be 𝑟.
The sequence is:
𝑎, 𝑎𝑟, 𝑎𝑟 2 , 𝑎𝑟 3 , …
Therefore,
𝑇1 = 𝑎
𝑇2 = 𝑎𝑟
𝑇3 = 𝑎𝑟 2
𝑇4 = 𝑎𝑟 3
For the 𝑛𝑡ℎ term, 𝑎is multiplied by 𝑟𝑛 − 1times.
Hence,
𝑇𝑛 = 𝑎𝑟 𝑛−1
3. Harmonic Sequence
A harmonic sequence is obtained when the reciprocals of its terms form an
arithmetic sequence.
Let the reciprocal sequence be:
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
Its general term is:
𝑎 + (𝑛 − 1)𝑑
Since the original sequence consists of reciprocals, we take the reciprocal:
𝑇𝑛 =
1
𝑎 + (𝑛 − 1)𝑑
𝑇𝑛 =
1
𝑎 + (𝑛 − 1)𝑑
Therefore,
Example
For:
1 1 1 1
, , , ,…
2 5 8 11
the reciprocal sequence is:
2,5,8,11, …
Here 𝑎 = 2and 𝑑 = 3.
Thus,
1
2 + (𝑛 − 1)3
1
𝑇𝑛 =
3𝑛 − 1
𝑇𝑛 =