Arithmetic Progressions

5.1 Introduction

This introductory section highlights the presence of patterns in nature and everyday life, which often follow specific sequences or progressions. It begins by pointing out natural examples such as the petals of a sunflower, the holes in a honeycomb, grains on a maize cob, and spirals on pineapples and pine cones, all exhibiting regular patterns. The section then transitions to patterns observed in daily life, emphasizing that many sequences of numbers or measurements follow a certain rule or pattern. Several examples are provided to illustrate this concept: (i) Reena's salary progression: She starts with a monthly salary of ₹8000 and receives an annual increment of ₹500. Her salary over the years forms the sequence 8000, 8500, 9000, ... (ii) Ladder rung lengths: The lengths decrease uniformly by 2 cm from bottom to top, starting with 45 cm at the bottom rung. The lengths are 45, 43, 41, 39, 37, 35, 33, 31 cm. (iii) Savings scheme: An investment of ₹8000 grows by a factor of 5/4 every 3 years, resulting in maturity amounts of 10000, 12500, 15625, 19531.25 after 3, 6, 9, and 12 years respectively. (iv) Number of unit squares in squares of side 1, 2, 3, ... units, which are 1², 2², 3², ... (v) Shakila's daughter’s money box: Starting with ₹100 on the first birthday and increasing by ₹50 each year, the amounts are 100, 150, 200, 250, ... (vi) Rabbit pairs reproduction: Starting with one pair, the number of pairs over months follows the sequence 1, 1, 2, 3, 5, 8, ... The section concludes by noting that these examples show different types of patterns: some involve adding a fixed number (arithmetic progression), some involve multiplication, and others involve squares or Fibonacci-type sequences. The chapter will focus on the pattern where succeeding terms are obtained by adding a fixed number to the preceding term, known as Arithmetic Progressions (AP). It will cover how to find the nth term and the sum of n terms of such sequences and apply these concepts to solve real-life problems. **Table on page 13 (6×5)** | | a | d | n | an | | --- | --- | --- | --- | --- | | (i) | 7 | 3 | 8 | ... | | (ii) | -18 | ... | 10 | 0 | | (iii) | ... | -3 | 18 | -5 | | (iv) | -18.9 | 2.5 | ... | 3.6 | | (v) | 3.5 | 0 | 105 | ... |

• Patterns exist in nature and daily life, often following specific sequences.
• Examples include salary increments, ladder rung lengths, savings growth, and biological reproduction.
• Some sequences involve adding a fixed number; others involve multiplication or squares.
• Arithmetic Progression (AP) is a sequence where each term is obtained by adding a fixed number to the previous term.
• The chapter focuses on understanding APs, finding nth terms, and sums of terms.
• Real-life applications of APs will be explored.
• 📌 Arithmetic Progression (AP): A sequence where each term after the first is obtained by adding a fixed number to the preceding term.
• 📌 Common difference: The fixed number added to each term to get the next term in an AP.

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.

• An AP is a sequence where each term after the first is obtained by adding a fixed number (common difference) to the previous term.
• The common difference d can be positive, negative, or zero.
• Notation: first term a1 = a, nth term an, common difference d.
• General form of AP: a, a + d, a + 2d, a + 3d, ...
• Finite APs have a last term; infinite APs do not.
• To verify an AP, check if differences between consecutive terms are equal.
• 📌 Arithmetic Progression (AP): Sequence with constant difference between consecutive terms.
• 📌 Common difference (d): Fixed number added to each term to get the next.
• 📌 Finite AP: AP with a last term.