Arithmetic Progression (AP) Overview
Arithmetic Progression (AP) is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is often referred to as the common difference.
General Form of an AP
The general form of an Arithmetic Progression is expressed as:
$$ a, a + d, a + 2d, a + 3d, \ldots $$
where \( a \) is the first term and \( d \) is the common difference.
nth Term of an AP
The nth term of an AP (also known as the general term) is given by:
$$ a_n = a + (n  1)d $$
where \( a_n \) is the nth term, and \( n \) is the term number.
Sum of the First n Terms of an AP
The sum of the first \( n \) terms of an AP is calculated as:
$$ S_n = \frac{n}{2} (2a + (n  1)d) $$
or
$$ S_n = \frac{n}{2} (a + a_n) $$
where \( S_n \) is the sum of the first \( n \) terms.
Average of the First n Terms of an AP
The average of the first \( n \) terms of an AP is determined by:
$$ \text{Average} = \frac{S_n}{n} $$
where \( S_n \) is the sum of the first \( n \) terms and \( n \) is the number of terms.
Geometric Progression (GP) Overview
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, nonzero number called the common ratio.
General Form of a GP
The general form of a Geometric Progression is expressed as:
$$ a, ar, ar^2, ar^3, \ldots $$
where \( a \) is the first term and \( r \) is the common ratio.
nth Term of a GP
The nth term of a GP is given by:
$$ a_n = a \cdot r^{(n1)} $$
where \( a_n \) is the nth term, and \( n \) is the term number.
Sum of the First n Terms of a GP
For \( r \neq 1 \), the sum of the first \( n \) terms of a GP is calculated as:
$$ S_n = \frac{a (1  r^n)}{1  r} $$
where \( S_n \) is the sum of the first \( n \) terms.
Sum of an Infinite GP
For \( r < 1 \), the sum of the infinite terms in a GP is:
$$ S_{\infty} = \frac{a}{1  r} $$
where \( S_{\infty} \) is the sum of the infinite terms in the GP.
Product of the First n Terms of a GP
The product of the first \( n \) terms of a GP is given by:
$$ P_n = a^n \cdot r^{n(n1)/2} $$
where \( P_n \) is the product of the first \( n \) terms.
Harmonic Progression (HP) Overview
A Harmonic Progression (HP) is a sequence of numbers formed by taking the reciprocals of an arithmetic progression. Each term of an HP is the reciprocal of the corresponding term in the AP.
General Form of an HP
The general form of a Harmonic Progression is expressed as:
$$ \frac{1}{a}, \frac{1}{a + d}, \frac{1}{a + 2d}, \ldots $$
where \( a \) is the first term and \( d \) is the common difference of the corresponding AP.
nth Term of an HP
The nth term of an HP is the reciprocal of the nth term of the corresponding AP:
$$ \frac{1}{a_n} = \frac{1}{a + (n  1)d} $$
Fibonacci Sequence Overview
The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.
Fibonacci Sequence Formula
The Fibonacci Sequence is typically started with the terms 0 and 1, and each subsequent term is the sum of the previous two:
$$ 0, 1, 1, 2, 3, 5, 8, 13, \ldots $$
Mathematically, it can be expressed as \( F_n = F_{n1} + F_{n2} \), with \( F_0 = 0 \) and \( F_1 = 1 \).
Simple Interest (SI) Overview
Simple Interest (SI) is a method of calculating the interest charge on a loan or financial product based on the original principal and a fixed interest rate. It is widely used in finance for its straightforward approach.
Simple Interest Formula
The formula to calculate Simple Interest is:
$$ \text{Simple Interest (SI)} = \frac{P \times R \times T}{100} $$
where:
 \( P \) is the principal amount (the initial sum of money borrowed or invested).
 \( R \) is the annual interest rate (in percentage).
 \( T \) is the time the money is borrowed or invested for, in years.
Total Amount after Interest
The total amount after interest is added to the principal is calculated as:
$$ A = P + \text{SI} $$
or
$$ A = P \left(1 + \frac{R \times T}{100}\right) $$
where \( A \) is the total amount after adding the interest to the principal.
To Find Principal, Rate, or Time if other quantities are known
To find one of the quantities (Principal, Rate, or Time) if the others are known:
 Principal \( P = \frac{\text{SI} \times 100}{R \times T} \)
 Rate \( R = \frac{\text{SI} \times 100}{P \times T} \)
 Time \( T = \frac{\text{SI} \times 100}{P \times R} \)
Compound Interest (CI) Overview
Compound Interest (CI) is the interest calculated on both the initial principal and the accumulated interest from previous periods. It plays a crucial role in understanding the growth of investments and loans over time.
Compound Interest Formula
The formula to calculate the future value of an investment or loan with compound interest is:
$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$
where:
 \( A \) is the future value of the investment/loan, including interest.
 \( P \) is the principal investment/loan amount.
 \( r \) is the annual interest rate (decimal).
 \( n \) is the number of times interest is compounded per year.
 \( t \) is the time the money is invested or borrowed for, in years.
Compound Interest Only
To find just the compound interest (excluding the principal), the formula is:
$$ \text{CI} = P \left(1 + \frac{r}{n}\right)^{nt}  P $$
or
$$ \text{CI} = A  P $$
To Solve for Principal, Rate, Time, or Compounding Frequency
To find one of the variables (Principal, Rate, Time, or Compounding Frequency) if others are known, the main formula can be rearranged:
 Principal \( P = \frac{A}{(1 + \frac{r}{n})^{nt}} \)
 Rate \( r = n \left[\left(\frac{A}{P}\right)^{\frac{1}{nt}}  1\right] \)
 Time \( t = \frac{1}{n} \cdot \frac{\log(\frac{A}{P})}{\log(1 + \frac{r}{n})} \)
 Compounding Frequency \( n = \frac{r}{\left[\left(\frac{A}{P}\right)^{\frac{1}{rt}}  1\right]} \)
The Concept of Average
The concept of average is a fundamental statistical measure, often used to find the central or typical value in a set of numbers. The most common type of average is the arithmetic mean.
Arithmetic Mean (Common Average)
The arithmetic mean is calculated as:
$$ \text{Average} = \frac{\text{Sum of all elements in the set}}{\text{Number of elements in the set}} $$
Example: For a set of numbers \( 2, 3, 5, 7 \), the average is \( \frac{17}{4} = 4.25 \).
Other Types of Averages
Median
The median is the middle number in a sorted list of numbers. If the list has an odd number of elements, the median is the middle one. If the list has an even number of elements, the median is the average of the two middle numbers.
Mode
The mode is the number that appears most frequently in a data set. A set of numbers may have one mode, more than one mode, or no mode at all.
Weighted Average
A weighted average is calculated by multiplying each component by a factor reflecting its importance and then summing these products.
$$ \text{Weighted Average} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}} $$
Geometric Mean
The geometric mean is calculated by multiplying all the numbers together and then taking the nth root (where n is the number of values). It is useful for sets of positive numbers.
Harmonic Mean
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.
$$ \text{Harmonic Mean} = \frac{n}{\sum (\frac{1}{\text{each number}})} $$
It is particularly useful in situations where the average of rates is desired.
Mensuration Overview
Mensuration is the branch of mathematics that deals with the measurement of various geometric figures and shapes, including their area, volume, and perimeter.
2D Shapes

Rectangle
Area: \( \text{Area} = \text{length} \times \text{width} \)
Perimeter: \( \text{Perimeter} = 2(\text{length} + \text{width}) \)

Square
Area: \( \text{Area} = \text{side}^2 \)
Perimeter: \( \text{Perimeter} = 4 \times \text{side} \)

Triangle
Area: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Perimeter: \( \text{Perimeter} = \text{side}_1 + \text{side}_2 + \text{side}_3 \)
For equilateral triangle, Area: \( \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 \)

Circle
Area: \( \text{Area} = \pi \times \text{radius}^2 \)
Circumference: \( \text{Circumference} = 2 \pi \times \text{radius} \)

Parallelogram
Area: \( \text{Area} = \text{base} \times \text{height} \)
Perimeter: \( \text{Perimeter} = 2(\text{side}_1 + \text{side}_2) \)

Trapezium
Area: \( \text{Area} = \frac{1}{2} (\text{base}_1 + \text{base}_2) \times \text{height} \)
Perimeter: Sum of all sides.

Rhombus
Area: \( \text{Area} = \frac{\text{diagonal}_1 \times \text{diagonal}_2}{2} \)
Perimeter: \( \text{Perimeter} = 4 \times \text{side} \)

Kite
Area: \( \text{Area} = \frac{\text{diagonal}_1 \times \text{diagonal}_2}{2} \)
Perimeter: Sum of all sides.

Ellipse
Area: \( \text{Area} = \pi \times \text{semimajor axis} \times \text{semiminor axis} \)
Perimeter: Approximated by \( \text{Perimeter} \approx \pi \left[ 3(\text{a} + \text{b})  \sqrt{(3\text{a} + \text{b})(\text{a} + 3\text{b})} \right] \) where \( \text{a} \) and \( \text{b} \) are the semimajor and semiminor axes.

Regular Pentagon
Area: \( \text{Area} = \frac{1}{4} \times \sqrt{5(5 + 2\sqrt{5})} \times \text{side}^2 \)
Perimeter: \( \text{Perimeter} = 5 \times \text{side} \)

Regular Hexagon
Area: \( \text{Area} = \frac{3}{2} \times \sqrt{3} \times \text{side}^2 \)
Perimeter: \( \text{Perimeter} = 6 \times \text{side} \)

Regular Octagon
Area: \( \text{Area} = 2(1 + \sqrt{2}) \times \text{side}^2 \)
Perimeter: \( \text{Perimeter} = 8 \times \text{side} \)
3D Shapes

Cube
Volume: \( \text{Volume} = \text{side}^3 \)
Surface Area: \( \text{Surface Area} = 6 \times \text{side}^2 \)

Cuboid
Volume: \( \text{Volume} = \text{length} \times \text{width} \times \text{height} \)
Surface Area: \( \text{Surface Area} = 2(\text{length} \times \text{width} + \text{width} \times \text{height} + \text{height} \times \text{length}) \)

Cylinder
Volume: \( \text{Volume} = \pi \times \text{radius}^2 \times \text{height} \)
Surface Area: \( \text{Surface Area} = 2\pi \times \text{radius} \times (\text{radius} + \text{height}) \)

Cone
Volume: \( \text{Volume} = \frac{1}{3} \pi \times \text{radius}^2 \times \text{height} \)
Surface Area: \( \text{Surface Area} = \pi \times \text{radius} \times (\text{radius} + \text{slant height}) \)

Sphere
Volume: \( \text{Volume} = \frac{4}{3} \pi \times \text{radius}^3 \)
Surface Area: \( \text{Surface Area} = 4 \pi \times \text{radius}^2 \)

Hemisphere
Volume: \( \text{Volume} = \frac{2}{3} \pi \times \text{radius}^3 \)
Surface Area: \( \text{Surface Area} = 3 \pi \times \text{radius}^2 \)

Prism
Volume: \( \text{Volume} = \text{Base Area} \times \text{height} \)
Surface Area: Depends on the shape of the base and includes the area of the base and the lateral surface area.

Pyramid
Volume: \( \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{height} \)
Surface Area: Sum of the base area and the lateral surface area.

Frustum of a Cone
Volume: \( \text{Volume} = \frac{1}{3} \pi \times \text{height} \times (\text{radius}_1^2 + \text{radius}_1 \times \text{radius}_2 + \text{radius}_2^2) \)
Surface Area: \( \text{Surface Area} = \pi \times (\text{radius}_1 + \text{radius}_2) \times \text{slant height} + \pi \times (\text{radius}_1^2 + \text{radius}_2^2) \)

Torus
Volume: \( \text{Volume} = 2 \pi^2 \times \text{radius}_1 \times \text{radius}_2^2 \)
Surface Area: \( \text{Surface Area} = 4 \pi^2 \times \text{radius}_1 \times \text{radius}_2 \)
Here, \( \text{radius}_1 \) is the distance from the center of the tube to the center of the torus, and \( \text{radius}_2 \) is the radius of the tube.