Geometric Progression Chapter Notes | Mathematics Class 10 ICSE PDF Download

Introduction

Dive into the fascinating world of Geometric Progression (G.P.), where numbers dance in a rhythmic pattern, each term multiplying or dividing its predecessor by a fixed quantity! This chapter unveils the magic of sequences and series that grow or shrink consistently, forming the backbone of many mathematical and real-world applications. From understanding how terms connect through a common ratio to exploring infinite sums and geometric means, this journey through G.P. will sharpen your problem-solving skills and spark curiosity about the patterns hidden in numbers. Let’s unravel the elegance of Geometric Progression together!

• A sequence is a list of numbers arranged in a specific order following a rule.
• Each number in the sequence is called a term.
• A series is formed by connecting the terms of a sequence with plus or minus signs.
• In this chapter, sequences and series are often considered the same.
• Example: The sequence 2, 6, 18, 54 forms a series when written as 2 + 6 + 18 + 54. Each term is multiplied by 3 to get the next term.

Geometric Progression

• A geometric progression (G.P.) is a sequence where each term is obtained by multiplying or dividing the previous term by a fixed number, called the common ratio (denoted by r).
• Abbreviated as G.P., it’s identified by checking if the ratio between consecutive terms is constant.
• The common ratio r is found by dividing any term by its preceding term.
• Formula: If a is the first term and r is the common ratio, the n^thterm of a G.P. is:
  t_n = a × r^n-1
• Example 1: For the sequence 2, 4, 8, 16:
  • Each term is multiplied by 2 to get the next term.
  • Common ratio r = 4/2 = 2.
  • Thus, it’s a G.P. with first term a = 2 and common ratio r = 2.
• Example 2: Find the 6^thterm of the G.P. with first term 125 and second term 25.
  • Given: First term a = 125, second term = ar = 25.
  • Common ratio r = 25/125 = 1/5.
  • 6^th term = ar⁵ = 125 × (1/5)⁵ = 125/3125 = 1/25.

General Term of a Geometric Progression

• The general term (or n^th term) of a G.P. helps find any term in the sequence without listing all previous terms.
• It is derived from the pattern where each term is the first term multiplied by the common ratio raised to the power of (term number - 1).
• Formula: For a G.P. with first term a and common ratio r, the n^thterm is:
  t_n = a × r^n-1
• Example 1: For the G.P. 12, 4, 4/3, find the 10^thterm.
  • First term a = 12, common ratio r = 4/12 = 1/3.
  • 10^th term = t₁₀ = 12 × (1/3)^10-1 = 12 × (1/3)⁹ = 12 / 19683 = 4/6561.
• Example 2: Find the 8^thterm of the sequence 3/4, 3/2, 3.
  • First term a = 3/4, common ratio r = (3/2) / (3/4) = 3/2 × 4/3 = 2.
  • 8^th term = t₈ = (3/4) × 2^8-1 = (3/4) × 2⁷ = (3/4) × 128 = 96.

Properties of Geometric Progression

• If each term of a G.P. is multiplied or divided by a non-zero constant, the resulting sequence is still a G.P. with the same common ratio.
• The reciprocals of the terms of a G.P. form another G.P. with common ratio 1/r.
• If a, b, c are in G.P., then b² = a × c.
• Example 1: If the 4^th, 7^th, and 10^th terms of a G.P. are a, b, and c, then b²= a × c.
  • Let first term = A, common ratio = R.
  • 4^th term = AR³ = a, 7^th term = AR⁶ = b, 10^th term = AR⁹ = c.
  • a × c = AR³ × AR⁹ = A²R¹² = (AR⁶)² = b².
• Example 2: If the 3^rdterm of a G.P. is 4, find the product of its first five terms.
  • Let first term = a, common ratio = r. Given: 3^rd term = ar² = 4.
  • Product of first five terms = a × ar × ar² × ar³ × ar⁴ = a⁵r¹⁰ = (ar²)⁵ = 4⁵ = 1024.

Sum of Finite Terms of a Geometric Progression

The sum of the first n terms of a G.P. depends on the common ratio r.

Formulas:

Alternative forms using last term l = ar^n-1:

Example 1: Find the sum of 10 terms of the series 96 - 48 + 24.

Alternative Method: For the same series, using the last term.

Example 2: Find the sum of 8 terms of the G.P. 3 + 6 + 12 + 24 + ...

Sum of Infinite Terms of a G.P.

• For a G.P. with first term a and common ratio r, if the absolute value of the common ratio is less than 1 (|r| < 1), the sum of infinite terms exists.
• Formula: Sum of infinite terms =
  S = a / (1 - r), where |r| < 1
• Example 1: Find the sum of the G.P. 1 - 1/4 + 1/16 - 1/64 + ... up to infinite terms.
  • First term a = 1, common ratio r = (-1/4)/1 = -1/4.
  • |r| = 1/4 < 1, so the sum exists.
  • Sum = a / (1 - r) = 1 / (1 - (-1/4)) = 1 / (1 + 1/4) = 4/5.
• Example 2: Find the first term of a G.P. with second term 2 and sum of infinite terms 8.
  • Given: ar = 2 and a / (1 - r) = 8, where |r| < 1.
  • From ar = 2, we get a = 2/r.
  • Substitute in sum formula: (2/r) / (1 - r) = 8.
  • Simplify: 2 = 8r - 8r² → 4r² - 4r + 1 = 0.
  • Solve: (2r - 1)² = 0 → r = 1/2.
  • Then, a = 2 / (1/2) = 4.
  • First term is 4.

 Recurring Decimals

• Recurring decimals can be expressed as the sum of an infinite G.P.
• Use the formula for the sum of an infinite G.P. to convert recurring decimals to fractions.
• Formula: For a recurring decimal, sum =
  S = a / (1 - r), where |r| < 1
• Example 1: Evaluate 0.4373737....
  • Write as: 0.437 = 0.4 + 0.037 + 0.00037 + 0.0000037 + ...
  • First term a = 37/1000, common ratio r = 1/100.
  • Sum of infinite terms = (37/1000) / (1 - 1/100) = (37/1000) / (99/100) = 37/990.
  • Add non-recurring part: 0.4 = 4/10.
  • Total = 4/10 + 37/990 = (396 + 37)/990 = 433/990.

Geometric Mean between Numbers a and b

• The geometric mean (G) between two positive numbers a and b forms a G.P.: a, G, b.
• Formula: Geometric mean =
  G = √(a × b)
• Example 1: Find the geometric mean between 3 and 12.
  • G = √(3 × 12) = √36 = 6.
• Example 2: Find the geometric mean between 3 and 243.
  • G = √(3 × 243) = √729 = 27.

n Geometric Means between a and b

• To insert n geometric means between a and b, form a G.P. with n + 2 terms: a, G₁, G₂, ..., G_n, b.
• The common ratio r is found using: b = a × rⁿ⁺¹.
• Each mean is calculated as: G_k = a × r^k.
  
• Example 1: Insert two numbers between 3 and 81 to form a G.P.
  • Sequence: 3, G₁, G₂, 81 (4 terms).
  • 81 = 3 × r³ → r³ = 27 → r = 3.
  • G₁ = 3 × 3 = 9, G₂ = 3 × 3² = 27.
  • Sequence: 3, 9, 27, 81.
• Example 2: Insert three numbers between 1 and 256 to form a G.P.
  • Sequence: 1, G₁, G₂, G₃, 256 (5 terms).
  • 256 = 1 × r⁴ → r⁴ = 256 = 4⁴ → r = 4.
  • G₁ = 1 × 4 = 4, G₂ = 1 × 4² = 16, G₃ = 1 × 4³ = 64.
  • Sequence: 1, 4, 16, 64, 256.