What is a Geometric Progression?

S3-SA1-0155

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 today. Tomorrow, you get 2 times that amount (2 Rupees). The day after, you get 2 times 2 Rupees (4 Rupees), and so on. The sequence 1, 2, 4, 8, 16... is a Geometric Progression because each number is found by multiplying the previous one by 2.

Worked Example

Step-by-Step

Let's find the 4th term of a GP where the first term is 3 and the common ratio is 2.

Step 1: Identify the first term (a) and the common ratio (r).
Here, a = 3 and r = 2.

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Step 2: Write out the terms using the definition.
The first term is 3.

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Step 3: To find the second term, multiply the first term by the common ratio.
Second term = 3 * 2 = 6.

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Step 4: To find the third term, multiply the second term by the common ratio.
Third term = 6 * 2 = 12.

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Step 5: To find the fourth term, multiply the third term by the common ratio.
Fourth term = 12 * 2 = 24.

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Answer: The 4th term of the GP is 24.

Why It Matters

Geometric Progressions are super useful for understanding how things grow or decay exponentially, like compound interest in banks or how populations grow. Engineers use GPs to design antennas, and data scientists use them to model growth patterns in data. Learning GPs opens doors to exciting careers in finance, technology, and science!

Common Mistakes

MISTAKE: Confusing GP with Arithmetic Progression (AP) and adding the common difference instead of multiplying. | CORRECTION: Remember, in a GP, you always multiply by a common ratio to get the next term. In an AP, you add or subtract a common difference.

MISTAKE: Calculating the common ratio by subtracting terms. | CORRECTION: The common ratio (r) is found by dividing any term by its previous term (e.g., r = Term2 / Term1 or r = Term3 / Term2).

MISTAKE: Forgetting that the common ratio can be a fraction or a negative number. | CORRECTION: The common ratio can be any non-zero number. If r is a fraction (e.g., 1/2), the terms will decrease. If r is negative, the terms will alternate in sign.

Practice Questions

Try It Yourself

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

QUESTION: If the first term of a GP is 2 and the common ratio is 3, what is the 3rd term? | ANSWER: 18

QUESTION: A bacterial colony doubles its size every hour. If it starts with 100 bacteria, how many bacteria will there be after 3 hours? | ANSWER: 800 (100 -> 200 -> 400 -> 800)

MCQ

Quick Quiz

Which of the following sequences is a Geometric Progression?

2, 4, 6, 8...

1, 3, 9, 27...

5, 10, 15, 20...

10, 8, 6, 4...

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. The common ratio is 3. Other options are Arithmetic Progressions or neither.

Real World Connection

In the Real World

Imagine you're tracking the spread of a rumour on social media. If each person tells two friends, and those two friends tell two more, the number of people hearing the rumour can grow in a Geometric Progression. This helps understand how information, or even viruses, can spread rapidly.

Key Vocabulary

Key Terms

Geometric Progression (GP): A sequence where each term after the first is found by multiplying by a fixed number | Common Ratio (r): The fixed non-zero number by which each term is multiplied to get the next term | Term: Each number in the sequence | Sequence: An ordered list of numbers

What's Next

What to Learn Next

Great job understanding Geometric Progressions! Next, you can explore how to find the sum of a GP, which is useful for calculating things like total savings over time. Keep practicing, and you'll master sequences and series!