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2.7 - Geometric Progression (G.P.)

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Introduction to Geometric Progression

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Teacher
Teacher

Welcome everyone! Today, we're going to explore Geometric Progression. A G.P. is a sequence where each term is obtained by multiplying the previous term by a constant number known as the common ratio. Can anyone give me an example of what a G.P. could look like?

Student 1
Student 1

Is 2, 6, 18 that a G.P.? I see each term is multiplied by 3.

Teacher
Teacher

Exactly! The first term is 2, and the common ratio, r, is 3. Great observation! Let's remember that the general form is \( a_n = a \cdot r^{n-1} \). Can anyone tell me the 2nd term in this sequence?

Student 2
Student 2

The second term would be 2 times 3, which is 6.

Teacher
Teacher

Correct! Now you’re thinking like mathematicians. The 3rd term is 18. So we see the pattern: multiply the first term by the common ratio each time. Let's summarize — we have a sequence, a common ratio, and we've found terms successfully!

Finding the nth Term

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Teacher
Teacher

Now let's delve deeper into how we can find the nth term of a G.P. Who can remind us of the formula?

Student 3
Student 3

Isn't it \( a_n = a \cdot r^{n-1} \)?

Teacher
Teacher

That's right! So if we have a G.P. with a first term of 2 and a common ratio of 3, what would the 5th term be?

Student 4
Student 4

It's \( 2 \cdot 3^{4} = 2 \cdot 81 = 162 \).

Teacher
Teacher

Correct again! You all are doing wonderfully. The 5th term is 162. If you need help remembering, think ‘Multiply the base to the power of one less than the term.’

Sum of the First n Terms

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Teacher
Teacher

Next, let's discuss how we calculate the sum of the first n terms in a G.P. Can anyone recall the formula for this?

Student 1
Student 1

It’s \( S_n = a \cdot \frac{r^n - 1}{r - 1} \)!

Teacher
Teacher

Excellent! Now let’s apply it. If we want to find the sum of the first 5 terms where a = 2 and r = 3, what will we get?

Student 2
Student 2

So \( S_5 = 2 \cdot \frac{3^5 - 1}{3 - 1} = 2 \cdot \frac{243 - 1}{2} = 2 \cdot 121 = 242 \).

Teacher
Teacher

Fantastic! You’ve just found the sum of the first 5 terms in this G.P. Remember, the sum is important when dealing with situations like financial savings or population growth. Great job, everyone!

Introduction & Overview

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Quick Overview

Geometric Progression involves sequences formed by multiplying each term by a constant ratio, with formulas governing the nth term and the sum of the series.

Standard

Geometric Progression (G.P.) is a crucial mathematical concept where each term in a sequence is produced by multiplying the previous term by a fixed non-zero number called the common ratio. This section covers the formula for the nth term and the sum of the first n terms, elucidating their significance in various mathematical applications.

Detailed

Geometric Progression (G.P.)

Geometric Progression, or G.P., is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (denoted as r). The first term of the sequence is represented by a. The general formulas for a G.P. include:

  • nth Term: The nth term of a G.P. can be written as:

\[ a_n = a \cdot r^{n-1} \]

where:
- a is the first term.
- r is the common ratio.
- n is the term's position in the sequence.

  • Sum of First n Terms: The sum of the first n terms (when r is not equal to 1) can be calculated using:

\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \]

This formula is essential for solving various problems involving sequences and series. Understanding G.P. is fundamental for deeper explorations in algebra and its applications in real-life contexts such as finance, population growth, and physics.

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Audio Book

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Definition of Geometric Progression

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● A sequence where each term is obtained by multiplying the previous term by a constant (common ratio).

Detailed Explanation

A Geometric Progression (G.P.) is a special type of sequence in mathematics where each term is derived from the previous term by multiplying it by a constant value known as the common ratio. This means if you have the first term of the sequence, you can easily calculate the next terms by multiplying the first term repeatedly by the common ratio.

Examples & Analogies

Imagine you have a plant that doubles its height every week. If your plant starts at 1 cm, the heights of your plant over weeks will form a geometric progression: 1 cm, 2 cm (1 * 2), 4 cm (2 * 2), 8 cm (4 * 2), and so on. Each week, you multiply the height of the previous week by 2.

Nth Term Formula

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● nth term: an=a⋅rn−1

Detailed Explanation

To find the nth term of a geometric progression, we use the formula an = a * r^(n-1), where 'an' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number you want to find. This allows you to compute any term in the sequence without needing to calculate every previous term.

Examples & Analogies

Consider investing money in a savings account that pays interest compounded annually. If you start with $100 (your first term) and earn 5% interest each year (the common ratio), you can find out how much money you will have after any number of years using the nth term formula. For example, after 3 years, you would calculate it as: a3 = 100 * 1.05^(3-1) = 100 * 1.1025 = $110.25.

Sum of First n Terms

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● Sum of first n terms (when r ≠ 1): Sn=a⋅(rn−1)/(r−1)

Detailed Explanation

The sum of the first n terms of a geometric progression can be calculated using the formula Sn = a * (r^n - 1) / (r - 1), where 'Sn' is the sum of the first n terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms you want to sum. This formula is particularly useful when the common ratio is not equal to 1.

Examples & Analogies

If you decide to double a certain amount of money every month, say you start with $10, you can find out how much money you have after 5 months using the sum formula. Here, it’s as if every month you’re adding more based on the previous month’s doubling, leading to a rapid increase in total savings.

Example of Finding the 5th Term

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● Example: Find the 5th term of a G.P. with a = 2, r = 3. Solution: a5=2×3^4=2×81=162.

Detailed Explanation

To find the 5th term of a geometric progression where the first term (a) is 2 and the common ratio (r) is 3, we use the nth term formula: a5 = 2 * 3^(5-1) = 2 * 3^4. First, calculate 3^4, which is 81, then multiply it by 2, resulting in a5 = 162. This shows how quickly values can grow in a G.P.

Examples & Analogies

Think of this example like a tree that triples its height every year. Starting with a height of 2 meters, after 4 years (to calculate the 5th height), the tree reaches 2 meters * 81 (which is the height achieved after 4 growths). This demonstrates the explosive growth one can expect in geometric patterns.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Geometric Progression: A sequence generated by multiplying the previous term by a fixed non-zero number.

  • Common Ratio: The constant factor used to multiply each term to obtain the next term in the sequence.

  • nth Term Formula: \( a_n = a \cdot r^{n-1} \), where a is the first term and r is the common ratio.

  • Sum of First n Terms Formula: \( S_n = a \cdot \frac{r^n - 1}{r - 1} \) for r ≠ 1.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: Given a = 2 and r = 3, the first five terms of the G.P. are 2, 6, 18, 54, 162.

  • Example 2: If you have a G.P. with a = 1 and r = 2, sum of the first four terms is S_4 = 1*(2^4-1)/(2-1) = 15.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • In a G.P., just multiply, For the next term, don't be shy!

📖 Fascinating Stories

  • A rabbit hops where each jump is twice the last, starting from one step, it leaps far and fast. Each jump grows exponentially, much like terms in a G.P.

🧠 Other Memory Gems

  • BIG r (for ‘base times the exponential growth’) helps you remember G.P. multiplication.

🎯 Super Acronyms

GAP

  • G.P.
  • A: for a term
  • P: for power of r.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Geometric Progression (G.P.)

    Definition:

    A sequence of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio.

  • Term: Common Ratio (r)

    Definition:

    A fixed, non-zero number by which each term in a geometric progression is multiplied to get the next term.

  • Term: nth Term

    Definition:

    The term at position n in a sequence, calculated using the formula \( a_n = a \cdot r^{n-1} \).

  • Term: Sum of a Sequence

    Definition:

    The total obtained when all the terms of a sequence are added together.