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 ↓


General Term of an AP


Arithmetic progression, commonly abbreviated as AP, is a sequence of numbers in which the difference between any two consecutive terms remains constant. This difference is called the "common difference". It is a fundamental concept in algebra.

Let's first look at what an arithmetic progression looks like:

Consider the series: 2, 4, 6, 8, 10, 12, ...

Here, the difference between any two consecutive terms is 2, and thus 2 is the common difference. This series can continue to infinity.

Understanding the general term

In an arithmetic progression, the n-th term of the sequence can be defined by a specific formula. This is called the "general term" of the sequence, and it helps us find any term of the series without listing all the terms. The general term of an AP is expressed mathematically as follows:

    T n = a + (n - 1) * d

Here:

  • T n is the nth term.
  • a is the first term of the sequence.
  • n is the number of terms.
  • d is the common difference.

This formula provides a simple method to determine any term of an arithmetic progression without calculating all the terms before it.

Example 1: Finding the common term

Let us examine the series 3, 7, 11, 15, 19, ….

  • The first term is a 3 .
  • The common difference d is 4.

We can substitute these values into our formula to get the general term:

    T n = 3 + (n - 1) * 4

On substituting different values of n we will get different terms in this sequence.

Example 2: Using the common term formula

Suppose you are given the task of finding the 10th term of the sequence 5, 8, 11, 14, 17, ...

  • Here, a = 5
  • Common difference d = 3

Using the common term formula, we can calculate the 10th term:

    T 10 = 5 + (10 - 1) * 3
    T 10 = 5 + 9 * 3
    T 10 = 5 + 27
    T 10 = 32

Hence the 10th term is 32.

Visual example

Consider the AP: 1, 3, 5, 7, 9, 11, ...

    1, 3, 5, 7, 9, 11, ..., t n

For this AP:

    a = 1, d = 2

Plugging into the general term formula:

    t n = 1 + (n - 1) * 2

This helps in calculating any number of terms easily.

Example 3: Finding specific terms

Suppose there is an AP: 10, 15, 20, ...

You want to find the 15th term:

  • The first term is a 10.
  • The common difference d is 5.

Use the general term formula:

    T 15 = 10 + (15 - 1) * 5
    T 15 = 10 + 14 * 5
    T 15 = 10 + 70
    T 15 = 80

Hence the 15th term is 80.

Example 4: Understanding negative series

Arithmetic progression can also include negative numbers:

For series: 20, 15, 10, 5, 0, ...

  • The first term is a 20.
  • Here, the common difference is d -5.

We want to find the 7th term:

    T 7 = 20 + (7 - 1) * (-5)
    T 7 = 20 + 6 * (-5)
    T 7 = 20 – 30
    t 7 = -10

Hence the 7th term is -10.

Common real life applications

Arithmetic progressions have many applications, from architecture to economics. Urban planners can use APs when calculating evenly spaced buildings or lamp posts. Economists can use them to predict economic growth rates over successive years.

The importance of understanding AP

Mastering arithmetic progressions gives you a foundational tool in mathematics. Understanding the general terminology allows students, professionals, and enthusiasts to handle sequences efficiently without repetitive calculations.

Example 5: Large term numbers

Let's work with larger n values. For the AP, 6, 12, 18, 24, ..., let's find the 50th term.

  • a = 6
  • d = 6

Use of the formula:

    T 50 = 6 + (50 - 1) * 6
    T 50 = 6 + 49 * 6
    T 50 = 6 + 294
    T 50 = 300

Hence the 50th term is 300.

Drawbacks of practical computation

While you can calculate using scratch paper for small numbers, large numbers in AP quickly become cumbersome. Automated calculations or formulas can make the task much easier.

Conclusion

Common terms in arithmetic progressions are a versatile, efficient mathematical tool used to predict future terms of the sequence without step-by-step derivation of each term. Educationally, it forms a bridge to broader mathematical theories, enhancing logical problem-solving and analytical skills.

As you continue to practice and explore, the application and simplicity of finding terms in arithmetic progressions will become second nature to you, and will prove useful in a variety of mathematical topics and beyond.


Grade 10 → 2.4.2


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Definition of Arithmetic Progression
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Sum of First n Terms of an AP

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General Term of an AP

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