What is Arithmetic Progressions Class 10: Definition & Examples

Definition of Arithmetic Progressions in Class 10 Maths

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is called the common difference and is denoted by $d$.

If the first term of the AP is $a$, then the sequence looks like:

$$a, \; a + d, \; a + 2d, \; a + 3d, \; \ldots$$

For example, the sequence $3, 7, 11, 15, \ldots$ is an AP where the common difference $d = 4$.

In Class 10 NCERT Mathematics, understanding this definition is the first step to solving problems related to sequences and series.

Key Formulas in Arithmetic Progressions for Class 10

Two main formulas are essential when working with Arithmetic Progressions:

1. Nth Term of an AP:

$$a_n = a + (n - 1)d$$

where

• $a_n$ = nth term,
• $a$ = first term,
• $d$ = common difference,
• $n$ = term number.

2. Sum of First n Terms of an AP:

$$S_n = \frac{n}{2} [2a + (n - 1)d]$$

Alternatively,

$$S_n = \frac{n}{2} (a + a_n)$$

These formulas help you find any term or the sum of terms in an AP quickly.

Example: Find the 10th term of the AP: $5, 8, 11, 14, \ldots$

Here, $a = 5$, $d = 3$, $n = 10$

$$a_{10} = 5 + (10 - 1) \times 3 = 5 + 27 = 32$$

How to Identify Arithmetic Progressions in Real-Life Problems

Arithmetic Progressions appear in many real-life situations such as:

• Daily savings increasing by a fixed amount
• Seats arranged in rows with equal increments
• Planting trees at regular intervals

To identify an AP:

• Check if the difference between consecutive terms is constant.
• If yes, the sequence is an AP.

Example: A student saves ₹10 on the first day, ₹15 on the second day, ₹20 on the third day, and so on. Is this an AP?

Differences: $15 - 10 = 5$, $20 - 15 = 5$ (constant)

Yes, this is an AP with $a = 10$ and $d = 5$.

Difference Between Arithmetic Progression and Other Sequences

Understanding how AP differs from other sequences helps in solving problems correctly.

This comparison helps Class 10 students avoid confusion during exams.

Solved Example: Finding the Sum of an Arithmetic Progression

Problem: Find the sum of the first 20 terms of the AP: $7, 10, 13, 16, \ldots$

Solution:

Given:

• First term, $a = 7$
• Common difference, $d = 3$
• Number of terms, $n = 20$

Step 1: Find the 20th term using $a_n = a + (n-1)d$

$$a_{20} = 7 + (20 - 1) \times 3 = 7 + 57 = 64$$

Step 2: Use sum formula

$$S_n = \frac{n}{2} (a + a_n) = \frac{20}{2} (7 + 64) = 10 \times 71 = 710$$

Answer: The sum of the first 20 terms is 710.

Tips to Master Arithmetic Progressions for Class 10 Exams

• Memorize key formulas for nth term and sum of n terms.
• Practice identifying the common difference quickly.
• Solve varied problems from NCERT textbooks and sample papers.
• Use step-by-step methods to avoid careless mistakes.
• Understand word problems by translating them into AP sequences.
• Revise regularly to strengthen concepts.

Following these tips will help you score well in the Arithmetic Progressions chapter.