Arithmetic Progressions Study Guide

ARITHMETIC PROGRESSIONS

5.1 Introduction

  • Observation in Nature: Certain patterns are observable in nature, including:
    • Petals of a sunflower
    • Holes of a honeycomb
    • Grains on a maize cob
    • Spirals on a pineapple and pine cone
  • Daily Life Patterns:
    • Example 1: Job Offer Scenario
    • Reena's starting monthly salary: `8000
    • Annual Increment: `500
    • Salaries for years:
      • Year 1: `8000
      • Year 2: `8500
      • Year 3: `9000
      • …
    • Example 2: Ladder Rungs
    • Lengths decrease uniformly by 2 cm from bottom to top
    • Bottom rung: 45 cm
    • Lengths of rungs: 45, 43, 41, …, 31 cm (total: 8 rungs)
    • Example 3: Savings Scheme
    • Investment: `8000
    • After 3 years: `10000
    • Subsequent years: 12500,15625, `19531.25
    • Example 4: Unit Squares in Squares
    • Number of squares for side lengths 1, 2, 3, …: 1, 4, 9, …
    • Example 5: Money Box Savings
    • Initial amount: 100, incremented by50 yearly
    • Contributions: 100, 150, 200, 250…
    • Example 6: Rabbit Population Growth
    • Growth pattern: 1, 1, 2, 3, 5, 8…

5.2 Arithmetic Progressions (AP)

  • Definition: A list of numbers is termed as an Arithmetic Progression (AP) if each term after the first is obtained by adding a constant called the common difference (d) to the preceding term.
  • Notation:
    • First term: a₁
    • Second term: a₂
    • nth term: aₙ
    • Common difference: d
  • Example Lists:
    • (i) 1, 2, 3, 4, …; Each term increases by 1.
    • (ii) 100, 70, 40, …; Each term decreases by 30.
    • (iii) -3, -2, -1, 0, …; Each term increases by 1.
    • (iv) 3, 3, 3, 3, …; No change, d = 0.
    • (v) -1.0, -1.5, …; Decreases by 0.5.
  • General Form of an AP: The general form of an arithmetic progression can be written as:
    • a, a + d, a + 2d, a + 3d, …
  • Finite vs Infinite AP:
    • Finite AP: A sequence with a final term.
    • Infinite AP: A sequence without a final term.

5.3 nth Term of an AP

  • Finding the nth Term: The nth term is given by the formula:
    • a_n = a + (n - 1)d
  • Example Discussions:
    • Example 1: Reena's Job Salary
    • Year 5 salary calculation: 8000 + (5 - 1) × 500 = 10000
    • Generalizing:
    • n-th year salary: [8000 + (n-1) × 500]

Example Solutions:

  1. Find 10th Term:

    • For AP: 2, 7, 12…
    • Given: a = 2, d = 5, n = 10
    • Calculation: a_{10} = 2 + (10 - 1)×5 = 47

  2. Identifying Terms in an AP: Specific examples of lists and their terms identified by their common difference.

5.4 Sum of First n Terms of an AP

  • Formula for Sum:
    • S_n = \frac{n}{2} (2a + (n - 1)d)
    • Alternate format: S = \frac{n}{2} (a + l) (where l is the last term)
  • Example Case:
    • Shakila's daughter's savings over birthdays is an arithmetic sequence:
    • Savings contributions: ##100, 150, …, up to 21st birthday.
    • To compute the total amount: Use the sum formula with calculated values.

Examples of Sum Calculations:

  • Find the sum of first 22 terms for various sequences and CAS:
    • Consider numerical cases in class exercises.

5.5 Summary

  • Key Points Reviewed:
    1. Definition of AP: Obtained by adding a fixed d.
    2. Identifying AP: Check consistent differences.
    3. nth Term: a_n = a + (n - 1)d
    4. Sum of n Terms: Use relevant formulas per context.

Note: If a, b, and c are in arithmetic progression, then the relationship b = \frac{(a + c)}{2} holds, indicating that b is the arithmetic mean of a and c.