Arithmetic Progressions | Class 10 Mathematics Notes

5.2 Arithmetic Progressions

This section formally introduces Arithmetic Progressions (APs) by examining several lists of numbers and identifying the pattern of a fixed difference between consecutive terms. It begins with examples:

(i) 1, 2, 3, 4, ... (ii) 100, 70, 40, 10, ... (iii) -3, -2, -1, 0, ... (iv) 3, 3, 3, 3, ... (v) -1.0, -1.5, -2.0, -2.5, ...

Each term in these sequences is called a term. The section asks how to find the next term in each list and observes the pattern or rule. It notes that in each list, the next term is obtained by adding a fixed number to the previous term: +1, -30, +1, 0, and -0.5 respectively.

Such sequences where the difference between consecutive terms is constant are called Arithmetic Progressions (AP). The fixed number added is called the common difference (d), which can be positive, negative, or zero.

Notation is introduced: the first term is a1, second term a2, nth term an, and common difference d. The defining property is a2 - a1 = a3 - a2 = ... = an - a(n-1) = d.

More examples of APs are given:

(a) Heights of students: 147, 148, 149, ..., 157 (b) Minimum temperatures: -3.1, -3.0, -2.9, ..., -2.5 (c) Loan balance after paying 5% monthly: 950, 900, 850, ..., 50 (d) Cash prizes for toppers: 200, 250, 300, ..., 750 (e) Monthly savings: 50, 100, 150, ..., 500

The general form of an AP is expressed as a, a + d, a + 2d, a + 3d, ..., where a is the first term and d the common difference.

The section distinguishes between finite APs (with a last term) and infinite APs (without a last term). It emphasizes that to define an AP, both the first term and common difference must be known.

Examples with various values of a and d are provided to illustrate positive, negative, zero, fractional, and decimal common differences.

The section also explains how to verify if a given list is an AP by checking if the difference between consecutive terms is constant. It stresses that the difference is found by subtracting the kth term from the (k+1)th term.

Examples demonstrate this verification process and identify sequences that are not APs, such as the Fibonacci sequence and sequences with irregular differences.

Overall, this section lays the foundation for understanding and identifying Arithmetic Progressions.

📊 Diagram: Fig. 5.1 shows ladder rungs decreasing uniformly; Fig. 5.2 illustrates unit squares in squares of increasing side length; Fig. 5.3 depicts pairs of rabbits over months.

🧪 Activity: Identify whether given sequences form an AP by calculating differences between consecutive terms.

🔗 Connection: Prepares for the derivation of the nth term formula of an AP in section 5.3.