Introduction to Arithmetic Progressions and Common Difference for CBSE Class 10

Patterns surround us in nature and mathematics: the petals on a sunflower, the rungs of a ladder, the periodic ticking of a clock, and the compounding of interest. In algebra, when a sequence of numbers increases or decreases by a constant step from one term to the next, it is called an Arithmetic Progression (AP).

In CBSE Class 10 Mathematics, Chapter 5 (Arithmetic Progressions) is one of the most practical and scoring topics. Understanding the foundational definitions of the first term and the common difference is the essential gateway to calculating general terms, sums of series, and solving real-world word problems.

What You Will Learn

• Definition of sequences, series, and progressions
• Formal mathematical definition of an Arithmetic Progression (AP)
• The First Term ($a$) and the Common Difference ($d$)
• Nature of the common difference: positive, negative, or zero
• The general algebraic form of an AP: $a, a + d, a + 2d, a + 3d, \dots$
• Finite vs. Infinite Arithmetic Progressions
• How to test whether a given numerical sequence forms an AP (especially radical expressions)
• Solved CBSE board examination problems and common traps

1. What is an Arithmetic Progression?

Definition of an Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first term.

The fixed number added to each successive term is called the common difference of the AP, denoted by the letter $d$. The initial starting term is called the first term, denoted by the letter $a$ (or $a_1$).

Mathematical Expression for Common Difference:

For a sequence $a_1, a_2, a_3, \dots, a_n$: $d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \dots = a_{k+1} - a_k$

$\text{Common Difference } d = \text{Any Term} - \text{Its Immediately Preceding Term}$

Important: <u>To find the common difference $d$, you must ALWAYS subtract the preceding term from the succeeding term ($a_{k+1} - a_k$). Never subtract the larger term from the smaller term arbitrarily!</u>

2. Nature of the Common Difference ($d$)

The common difference $d$ can be positive, negative, or zero:

                         Nature of Common Difference (d)
                                        |
       +--------------------------------+--------------------------------+
       |                                |                                |
     d > 0                            d < 0                            d = 0
Increasing AP                    Decreasing AP                    Constant AP
(e.g., 2, 5, 8, 11, ...)         (e.g., 100, 70, 40, 10, ...)     (e.g., 7, 7, 7, 7, ...)
Terms grow larger                Terms grow smaller               All terms identical

1. Positive Common Difference ($d > 0$):
   • The terms increase as the sequence advances.
   • Example: $2, 5, 8, 11, 14, \dots$ (Here $a = 2, d = 5 - 2 = +3$).
2. Negative Common Difference ($d < 0$):
   • The terms decrease as the sequence advances.
   • Example: $100, 70, 40, 10, -20, \dots$ (Here $a = 100, d = 70 - 100 = -30$).
3. Zero Common Difference ($d = 0$):
   • All terms in the sequence are identical.
   • Example: $7, 7, 7, 7, \dots$ (Here $a = 7, d = 7 - 7 = 0$).

3. General Form of an AP

Let the first term of an AP be $a$ and the common difference be $d$. The terms of the AP can be expressed as:

$\text{1st term } a_1 = a$ $\text{2nd term } a_2 = a + d$ $\text{3rd term } a_3 = a + 2d$ $\text{4th term } a_4 = a + 3d$ $\vdots$ $\text{General Form: } a, \; a + d, \; a + 2d, \; a + 3d, \; \dots$

Finite vs. Infinite APs:

1. Finite AP: An AP that contains a limited, countable number of terms. It possesses a definite last term ($l$).
   • Example: Heights of 20 students in a class arranged in ascending order.
2. Infinite AP: An AP that continues endlessly without a terminating term.
   • Example: $1, 3, 5, 7, \dots$ (the sequence of all odd positive integers).

4. How to Check if a Given Sequence is an AP

To verify whether a sequence forms an AP, calculate the successive differences $a_2 - a_1, a_3 - a_2, a_4 - a_3$:

• If $a_{k+1} - a_k$ is the same constant for all values of $k$, the sequence is an AP.
• If even one difference deviates, the sequence is not an AP.

Solved Example 1: Radical Sequences (CBSE Board Classic)

Problem: Check whether the sequence $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \dots$ forms an AP. If it does, find its common difference and write the next two terms.

Solution:

1. Simplify each radical term by extracting square factors:
   • $a_1 = \sqrt{2}$
   • $a_2 = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
   • $a_3 = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
   • $a_4 = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$
2. Compute successive differences: $a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = \sqrt{2}$ $a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}$ $a_4 - a_3 = 4\sqrt{2} - 3\sqrt{2} = \sqrt{2}$
3. Since $a_{k+1} - a_k = \sqrt{2}$ is constant throughout, <u>the given sequence is an AP with common difference $d = \sqrt{2}$</u>.
4. Next Two Terms:
   • $a_5 = a_4 + d = 4\sqrt{2} + \sqrt{2} = 5\sqrt{2} = \sqrt{25 \times 2} = \sqrt{50}$
   • $a_6 = a_5 + d = 5\sqrt{2} + \sqrt{2} = 6\sqrt{2} = \sqrt{36 \times 2} = \sqrt{72}$

Solved Example 2: Negative and Fractional Differences

Problem: Write the first four terms of an AP when the first term is $a = -1.25$ and the common difference is $d = -0.25$.

Solution:

• $a_1 = a = -1.25$
• $a_2 = a_1 + d = -1.25 + (-0.25) = -1.50$
• $a_3 = a_2 + d = -1.50 + (-0.25) = -1.75$
• $a_4 = a_3 + d = -1.75 + (-0.25) = -2.00$
• The required first four terms are: $-1.25, -1.50, -1.75, -2.00$.

5. Summary and Examination Tips

Exam Tip: When simplifying sequences with radicals like $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \dots$, notice that $\sqrt{6} - \sqrt{3} \ne \sqrt{3}$. This sequence is NOT an AP! Do not assume radical sequences are APs without simplifying terms first.

Common Mistake: Calculating $d$ as $a_1 - a_2$ instead of $a_2 - a_1$. In decreasing sequences like $10, 6, 2, \dots$, subtracting incorrectly gives $d = +4$ instead of the true common difference $d = -4$.