Arithmetic Progressions | Class 10 Mathematics Notes

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.

📊 Diagram: Fig. 5.1; Fig. 5.2; Fig. 5.3 illustrating ladder rungs, squares with unit squares, and rabbit pairs respectively.

🧪 Activity: Observation and identification of patterns in natural objects and daily life sequences to motivate the study of Arithmetic Progressions.

🔗 Connection: Leads to the formal definition and identification of Arithmetic Progressions in section 5.2.

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