Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Arithmetic Progressions ↓


Definition of Arithmetic Progression


An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two successive numbers (terms) is constant. This fixed difference is known as the "common difference". Understanding it allows us to explore various properties and characteristics of this type of sequence in mathematics.

Understanding the terms

Let's break down the components of an arithmetic progression. First, we have:

  • First term (a): It is the starting number from which the sequence starts.
  • Common Difference (d): It is the difference that is added to each term to reach the next term in the sequence.
  • n-th term (Tn): The n-th term of an arithmetic sequence can be found using the following formula:
tn = a + (n - 1) * d

Example of arithmetic progression

To understand Arithmetic Progression better, let us consider a simple example:

Imagine we have this sequence: 3, 6, 9, 12, 15,...

  • Here, the first term is (a) 3.
  • The common difference (d) is 3, since each number is 3 units more than the previous number (6 - 3 = 3, 9 - 6 = 3, and so on).

Thus, the sequence can be defined as:

tn = 3 + (n - 1) * 3

Now, if we want to find the 5th term in the sequence, we substitute 5 for n:

T5 = 3 + (5 - 1) * 3
   = 3 + 4 * 3
   = 3 + 12
   = 15

Therefore, the 5th term is 15, which confirms the sequence we wrote at the beginning.

Visual understanding

To visualize this, consider the following:

3 6 9 12 15

These lines represent the common spacing between numbers, helping us to see the uniform growth of the sequence.

More examples

Let's look at some additional examples to deepen our understanding:

Example 1

Consider the sequence: 10, 15, 20, 25, ...

  • The first term, a, is 10.
  • Each term increases by 5, so d is 5.

So, the formula for the nth term is:

TN = 10 + (N - 1) * 5

If we want to find the 7th term:

T7 = 10 + (7 - 1) * 5
   = 10 + 30
   = 40

Thus, the 7th term is 40.

Example 2

What if the sequence is decreasing, like: 20, 17, 14, 11, ...?

  • Here, the first term is a 20.
  • The terms decrease by 3, so d is -3.

The formula for the n-th term is:

TN = 20 + (N - 1) * (-3)

Let us find the fourth term:

T4 = 20 + (4 - 1) * (-3)
   = 20 - 9
   = 11

So, as expected, the fourth term is 11.

Sum of an arithmetic progression

The sum of all the terms in an arithmetic progression can also be easily calculated. This summation is especially useful when you want to add a large number of terms without having to do it manually.

The formula for finding the sum of the first n terms (Sn) is:

Sn = n/2 * (2a + (n - 1) * d)

Let's take a sequence and find its sum:

Consider the sequence: 5, 10, 15, ..., where we want to find the sum of the first 6 terms.

Use the given formula:

a = 5
d = 5
N = 6

SN = 6/2 * (2 * 5 + (6 - 1) * 5)
   = 3 * (10 + 25)
   = 3 * 35
   = 105

Therefore, the sum of the first 6 terms is 105.

Arithmetic progression in real life

Arithmetic progressions are not just theoretical constructs; they often occur in real life. Any situation where the rate of increase or decrease is constant can be modeled using an arithmetic progression.

A great example of this is savings growth. If a person saves a certain amount every month, the total savings form an arithmetic sequence, with the monthly sum being the common difference.

Conclusion

Arithmetic progressions serve as fundamental concepts in algebra and form the basis of more sophisticated applications in both mathematics and its real-world applications. By mastering this concept, we gain insight into various natural and financial progressions that span across many fields of study.

The ease with which arithmetic progressions can be calculated makes them a powerful tool for both students and mathematicians, encouraging further exploration of mathematical sequences and categories.


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Definition of Arithmetic Progression

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