What is a Geometric Progression (GP)?

S3-SA1-0426

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Think of it as a pattern where numbers grow or shrink by multiplying by the same amount each time.

Simple Example

Quick Example

Imagine you have 1 Rupee. Every day, you double your money. On Day 1, you have 1 Rupee. On Day 2, you have 2 Rupees (1x2). On Day 3, you have 4 Rupees (2x2). This sequence (1, 2, 4, 8, 16...) is a Geometric Progression because you multiply by 2 each time.

Worked Example

Step-by-Step

Let's find the next three terms in the GP: 3, 6, 12, ...

Step 1: Identify the first term (a) and find the common ratio (r).
The first term (a) is 3.
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Step 2: Calculate the common ratio (r) by dividing any term by its previous term. Let's divide the second term by the first: r = 6 / 3 = 2.
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Step 3: Confirm the common ratio with another pair. r = 12 / 6 = 2. So, the common ratio is indeed 2.
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Step 4: To find the next term after 12, multiply 12 by the common ratio.
Next term = 12 * 2 = 24.
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Step 5: To find the term after 24, multiply 24 by the common ratio.
Next term = 24 * 2 = 48.
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Step 6: To find the term after 48, multiply 48 by the common ratio.
Next term = 48 * 2 = 96.

Answer: The next three terms in the GP are 24, 48, and 96.

Why It Matters

Geometric Progressions are super useful! They help in understanding how money grows with compound interest in banks, how populations increase, or even how computer algorithms like binary search work. Knowing GPs can open doors to careers in finance, data science, and even game development!

Common Mistakes

MISTAKE: Confusing GP with Arithmetic Progression (AP) and adding instead of multiplying to find the next term. | CORRECTION: In a GP, you always MULTIPLY by a common ratio to get the next term, not add.

MISTAKE: Calculating the common ratio by subtracting terms instead of dividing. | CORRECTION: The common ratio (r) is found by dividing a term by its preceding term (e.g., term2 / term1).

MISTAKE: Assuming the common ratio is always a whole number. | CORRECTION: The common ratio can be a fraction (e.g., 1/2), a decimal (e.g., 0.5), or even a negative number.

Practice Questions

Try It Yourself

QUESTION: What is the common ratio of the GP: 5, 10, 20, 40, ...? | ANSWER: 2

QUESTION: Find the next term in the GP: 81, 27, 9, ... | ANSWER: 3

QUESTION: If the first term of a GP is 2 and the common ratio is 3, what are the first five terms of the GP? | ANSWER: 2, 6, 18, 54, 162

MCQ

Quick Quiz

Which of these sequences is a Geometric Progression?

2, 4, 6, 8, ...

1, 3, 9, 27, ...

10, 8, 6, 4, ...

5, 10, 15, 20, ...

The Correct Answer Is:

Option B (1, 3, 9, 27, ...) is a GP because each term is multiplied by 3 to get the next term (common ratio = 3). The other options are Arithmetic Progressions where a fixed number is added or subtracted.

Real World Connection

In the Real World

Geometric Progressions are used in many places! For example, when you share a funny video on WhatsApp with 2 friends, and each of those friends shares it with 2 more, and so on, the number of people who see it grows in a GP. Also, in banking, compound interest calculations follow a GP, showing how your savings can grow exponentially over time.

Key Vocabulary

Key Terms

COMMON RATIO: The fixed non-zero number by which each term is multiplied to get the next term in a GP. | TERM: Each number in the sequence. | SEQUENCE: An ordered list of numbers. | PROGRESSION: A sequence with a specific pattern.

What's Next

What to Learn Next

Great job understanding GPs! Next, you can explore how to find the nth term of a GP using a formula, which will help you find any term without listing them all. This builds directly on finding the common ratio and will make solving GP problems even faster!