Frequently Asked Concepts and Formulas: SEA-AP003 - Aptitude Development Seminar-3

Frequently Asked Concepts and Formulas

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, non-zero 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:

$$ a_n = a \cdot r^{(n-1)} $$

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:

$$ P_n = a^n \cdot r^{n(n-1)/2} $$

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 \( F_n = F_{n-1} + F_{n-2} \), 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: \( \text{Area} = \pi \times \text{semi-major axis} \times \text{semi-minor axis} \)

    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 semi-major and semi-minor 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.