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 ↓


Sum of First n Terms of an AP


An Arithmetic Progression (AP) is a sequence of numbers where the difference of any two successive members is a constant. This constant difference, usually represented by the letter d, is called the common difference.

For example, the sequence 2, 4, 6, 8, 10 is an arithmetic progression because the difference between successive terms is always 2. Another sequence is 5, 10, 15, 20, where the common difference is 5.

Elements of arithmetic progression

Arithmetic progression is represented as:

A, A+D, A+2D, A+3D, ..., A+(n-1)D
  • a is the first term.
  • d is the common difference.
  • n is the number of terms.

The sum of the first n terms of an arithmetic progression

To find the sum of the first n terms of an AP, we use the following formula:

S n = n/2 × (2a + (n − 1) × d)

or equivalent

S n = n/2 × (a + l)

Where:

  • S n is the sum of the first n terms.
  • l is the last term of the sequence. Thus, l = a + (n-1) × d.

Derivation of the formula

Let us derive this formula with a simple logical interpretation. Consider a sequence of n terms:

a, (a + d), (a + 2d), ..., [a + (n - 1)d]

Now, if we write the sequence in reverse order:

[a + (n - 1)d], [a + (n - 2)d], ..., (a + d), a

Adding these sequences term by term, we get the pairs:

(a + [a + (n - 1)d]), ((a + d) + [a + (n - 2)d]), ..., ([a + (n - 1)d] + a)

Each pair has the same sum:

2a + (n - 1)d

There are n such pairs, so the total becomes:

n × [(2a + (n – 1)d) / 2] = n/2 × (2a + (n – 1)d)

Example 1

Consider the sequence 2, 5, 8, 11, 14 Find the sum of these first 5 terms.

Here, a = 2, d = 3, and n = 5.

S 5 = 5/2 × (2 × 2 + (5 - 1) × 3)
S 5 = 5/2 × (4 + 12)
S 5 = 5/2 × 16 = 5 × 8 = 40

Hence, the sum of the first 5 terms is 40.

Visual example

2 5 8 11 14

Example 2

Let us find the sum of the first 10 terms of the arithmetic series 3, 7, 11, 15...

Here, a = 3 and d = 4.

First, find the 10th term:

L = a + (n − 1) × d = 3 + (10 − 1) × 4 = 3 + 36 = 39

Now, use the sum formula:

S 10 = 10/2 × (3 + 39)
S 10 = 5 × 42 = 210

Therefore, the sum of the first 10 terms is 210.

Example 3: AP with negative difference

Consider an arithmetic progression where a = 20, d = -3, and n = 6.

Sequence: 20, 17, 14, 11, 8, 5.

Calculate the totals:

S 6 = 6/2 × (2 × 20 + (6 - 1) × (-3))
S 6 = 3 × (40 – 15)
S 6 = 3 × 25 = 75

Hence, the sum of the first 6 terms is 75.

Generalization

The formula S n = n/2 × (2a + (n - 1) × d) is universally applicable to all arithmetic progressions - whether the common difference is positive, negative, or zero.

Mathematical explorations

This method of adding terms in an arithmetic progression illustrates an important concept: the balancing of increasing and decreasing numbers to get a total. In arithmetic progressions, this occurs because the progressive addition of the common difference creates a symmetric operation from both ends of the progression.

Practical applications

Understanding the sum of an arithmetic progression is useful in real life where situations proceed in a linear manner, such as calculating the total amount of equally spaced installments, or estimating the amount of savings over a period of time.


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General Term of an AP
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