What is Arithmetic Progression Class 10: Definition & Examples

Definition of Arithmetic Progression in Class 10 Mathematics

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 usually denoted by $d$.

For example, the sequence:

$$2, 5, 8, 11, 14, \dots$$

is an AP because the difference between consecutive terms is $3$.

Key points:

• The first term is denoted by $a$.
• The $n^{th}$ term is denoted by $a_n$.
• Common difference $d = a_2 - a_1 = a_3 - a_2 = \dots$

This concept is fundamental in Class 10 NCERT Mathematics and helps solve many real-life and exam problems.

Formula for the $n^{th}$ Term of an Arithmetic Progression

To find any term in an AP, we use the formula:

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

where:

• $a_n$ = $n^{th}$ term,
• $a$ = first term,
• $d$ = common difference,
• $n$ = term number.

Example: Find the 10th term of the AP: $3, 7, 11, 15, \dots$

• Here, $a = 3$, $d = 4$, $n = 10$

Calculate:

$$a_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39$$

So, the 10th term is $39$.

Sum of the First $n$ Terms of an Arithmetic Progression

The sum of the first $n$ terms of an AP is given by:

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

Alternatively, if the last term $a_n$ is known:

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

Example: Find the sum of the first 5 terms of the AP: $4, 9, 14, 19, 24, \dots$

• $a = 4$, $d = 5$, $n = 5$

Calculate the 5th term:

$$a_5 = 4 + (5 - 1) \times 5 = 4 + 20 = 24$$

Calculate the sum:

$$S_5 = \frac{5}{2} (4 + 24) = \frac{5}{2} \times 28 = 5 \times 14 = 70$$

So, the sum is $70$.

Difference Between Arithmetic Progression and Other Sequences

Understanding how AP differs from other sequences helps clarify its properties.

This comparison helps students identify and solve problems correctly in Class 10 NCERT exercises.

How to Identify an Arithmetic Progression in Problems

To check if a sequence is an AP:

• Calculate the difference between consecutive terms.
• If the difference is constant, it is an AP.

Example: Check if the sequence $7, 10, 13, 16, 20$ is an AP.

Calculate differences:

• $10 - 7 = 3$
• $13 - 10 = 3$
• $16 - 13 = 3$
• $20 - 16 = 4$

Since the last difference is not $3$, the sequence is not an AP.

This method is useful for quickly verifying sequences in Class 10 NCERT problems.

Solved Example: Using Arithmetic Progression Formulas

Problem: The first term of an AP is 5 and the common difference is 2. Find:

1. The 15th term 2. The sum of the first 15 terms

Solution:

Given: $a = 5$, $d = 2$, $n = 15$

1. Find the 15th term:

$$a_{15} = a + (n - 1)d = 5 + 14 \times 2 = 5 + 28 = 33$$

2. Find the sum of first 15 terms:

$$S_{15} = \frac{15}{2} [2 \times 5 + (15 - 1) \times 2] = \frac{15}{2} [10 + 28] = \frac{15}{2} \times 38 = 15 \times 19 = 285$$

Answer: The 15th term is 33 and the sum is 285.