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, denoted by $d$.

For example, the sequence 2, 5, 8, 11, 14, ... is an AP because the difference between consecutive terms is 3.

Key points:

• The first term of an AP is denoted by $a$.
• The common difference $d$ can be positive, negative, or zero.

Understanding this definition is essential for solving problems in the NCERT Class 10 Arithmetic Progressions chapter.

General Term Formula of Arithmetic Progression

The general term or the $n^{th}$ term of an AP is given by 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, ...

• Here, $a = 3$, $d = 4$
• Using the formula:

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

So, the 10th term is 39.

Sum of First n Terms of an Arithmetic Progression

The sum of the first $n$ terms of an AP is important for many Class 10 problems. The formula is:

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

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

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

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

• Here, $a = 4$, $d = 5$, $n = 5$
• Using the formula:

$$S_5 = \frac{5}{2} [2 \times 4 + (5 - 1) \times 5] = \frac{5}{2} [8 + 20] = \frac{5}{2} \times 28 = 70$$

So, the sum is 70.

Identifying Arithmetic Progression: Key Characteristics

To check if a sequence is an AP, verify whether the difference between consecutive terms is constant.

If the difference is not constant, the sequence is not an arithmetic progression.

Applications of Arithmetic Progression in Class 10 Maths

Arithmetic Progressions have many practical applications in Class 10 Mathematics and real life:

• Calculating total savings when saving a fixed amount regularly
• Finding terms in number patterns
• Solving problems involving consecutive integers
• Determining distances covered with uniform speed increments

Mastering AP helps in solving NCERT exercises and CBSE exam questions confidently. Practice with solved examples and previous year questions to strengthen your skills.

Common Mistakes to Avoid While Solving AP Problems

Students often make these mistakes in AP problems:

• Confusing the common difference $d$ with the first term $a$
• Forgetting to use $(n-1)$ in the general term formula
• Mixing up sum formulas
• Not verifying if a sequence is truly an AP before applying formulas

Always write down given data clearly and double-check calculations to avoid errors.