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Interactive Audio Lesson
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Finding the 6th Term
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Today, let's talk about finding terms in a geometric sequence. For example, if we have the sequence 2, 6, 18, 54, what do you think the 6th term will be?
Is it possible to use the common ratio to find it, like we learned?
Exactly! The common ratio here is 3. So to find the 6th term, we use the formula T = ar^(n-1). Can someone calculate this?
So T = 2 * 3^(6-1) = 2 * 3^5 = 486?
Great job! Remember, whenever you're unsure, write out the formula and substitute carefully. By using the method we discussed, you can find any term in a sequence.
Sum of the First n Terms
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Now let's discuss how to find the sum of the first 7 terms of the sequence 1, 1/2, 1/4, ... What do we need?
We will use the sum formula! S = a * (1 - r^n) / (1 - r).
Exactly! Let's plug in the values. Here a = 1, r = 1/2, and n = 7. Can anyone calculate the sum?
So, S = 1 * (1 - (1/2)^7) / (1 - 1/2) = 1 * (1 - 1/128) / (1/2). This simplifies to 2 * (127/128) = 2 * 0.9921875 = 1.984375.
Well done! This shows how even small fractions can still add up to a significant amount when compounded properly.
Determining if a Sequence is Geometric
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Let's evaluate the sequence 2, 5, 10, 17. Is this geometric? How could we determine that?
We can find the ratio of consecutive terms. If it's always the same, then it's geometric.
The ratios are 5/2, 10/5, and 17/10. They're not the same!
Yes! Well done! Thus, this sequence is not geometric. Remember, the key idea is that the ratio must remain constant across all terms.
Real-World Application
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A real-life application: Bacteria triple every hour. If we start with 200, what will the population be after 5 hours?
We could use the formula A = a * r^n, right?
Exactly! With a = 200 and r = 3, how would you set it up?
So, A = 200 * 3^5 = 200 * 243 = 48600. So there will be 48,600 bacteria after 5 hours?
Perfect! This is a fantastic example of how geometry sequences apply to exponential growth in nature.
Introduction & Overview
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Quick Overview
Standard
The section provides various practice problems related to geometric sequences, including finding specific terms, sums of terms, and identifying geometric sequences. These problems encourage students to apply the concepts learned in the chapter about geometric sequences and series.
Detailed
In this section, we delve into practice problems that allow students to apply their knowledge of geometric sequences. The problems encompass different scenarios: calculating specific terms in sequences, finding the sum of the first n terms, identifying if a sequence is geometric, and solving real-world problems related to growth and decay. The emphasis is on ensuring students can translate the theoretical concepts learned earlier into practical application. Completing these practice problems will help solidify their understanding of geometric sequences, their formulas, and their real-life relevance.
Audio Book
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Problem 1: Finding the 6th term in a Sequence
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- Find the 6th term of the sequence: 2, 6, 18, 54,…
Detailed Explanation
To find the 6th term of a geometric sequence, we first need to identify the first term (𝑎) and the common ratio (𝑟). In this case, the first term is 2. We can find the common ratio by dividing the second term by the first term: 6 / 2 = 3. Thus, each term is obtained by multiplying the previous term by 3. We can find the 6th term using the formula: T = a * r^(n-1). Therefore, T = 2 * 3^(6-1) = 2 * 243 = 486.
Examples & Analogies
Think of a plant that triples its height every week. If it starts at 2 cm in the first week, after 6 weeks, its height can be calculated using the same rules as the 6th term of the sequence, giving us a tangible connection to how growth works over time.
Problem 2: Sum of the First 7 Terms
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- Determine the sum of the first 7 terms: 1, 1/2, 1/4,…
Detailed Explanation
To determine the sum of the first 7 terms in a geometric sequence, we need to identify the first term (𝑎) and the common ratio (𝑟). Here, the first term is 1 and the common ratio is 1/2, since each term is half of the previous one. We then use the sum formula for finite geometric series: S = a * (1 - r^n) / (1 - r), where n is the number of terms. Thus, S = 1 * (1 - (1/2)^7) / (1 - 1/2). After calculation, we find the sum.
Examples & Analogies
Imagine you are sharing a pizza with 7 friends, and with each sharing round, you take half of the remaining pizza each time. Calculating how much pizza you’ve eaten after 7 rounds can help students visualize the diminishing returns and sums of geometric sequences.
Problem 3: Checking for Geometric Sequence
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- Is the sequence 2, 5, 10, 17 geometric? Justify.
Detailed Explanation
To determine if a sequence is geometric, we need to check if the ratio of successive terms is constant. We calculate the ratios: 5/2, 10/5, and 17/10. If these ratios are the same, the sequence is geometric. In this case, the ratios are not equal (2.5, 2, 1.7), indicating that this is not a geometric sequence.
Examples & Analogies
It's like checking if a set of stairs is uniform. If each step is the same height (constant ratio), then they are part of a uniform series; if not, they represent different heights—just like this sequence.
Problem 4: Sum of the Infinite Series
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- If 𝑎 = 4, 𝑟 = 1/4, find the sum of the infinite series.
Detailed Explanation
To find the sum of an infinite geometric series, we need the first term (𝑎) and the common ratio (𝑟). In this case, 𝑎 is 4 and 𝑟 is 1/4. We use the formula for an infinite series: S = a / (1 - r), applicable only if |𝑟| < 1, which it is here. We calculate S = 4 / (1 - 1/4) = 4 / (3/4) = 4 * (4/3) = 16/3.
Examples & Analogies
This can be likened to money being added to your savings account every month, where the first deposit is $4 and each month you add a quarter of the remaining previous balance. Even if the amounts reduce, they form a sum that can be calculated using our geometric series formula.
Problem 5: Exponential Growth in Bacteria Population
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- The population of bacteria triples every hour. Starting from 200, what is the population after 5 hours?
Detailed Explanation
In this problem, we need to find the population after 5 hours, knowing that the bacteria triple every hour. The first term is 200, and the common ratio is 3. We can use the formula for the n-th term: T = a * r^(n-1). So for 5 hours, T = 200 * 3^(5) = 200 * 243 = 48600. Thus, after 5 hours, the bacteria population grows to 48,600.
Examples & Analogies
Imagine a video game where every hour, your character's health triples. Starting from 200 health points, how powerful would your character be after 5 hours? This helps students visualize exponential growth as they relate to progress in gaming.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Geometric Sequence: A sequence formed by multiplying a starting number by a constant ratio.
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Common Ratio: A fixed value that connects terms in a geometric sequence.
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Finite Series: A series that includes a limited number of terms.
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Infinite Series: A series with an infinite number of terms that converges under specific conditions.
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Real-world Applications: Geometric sequences are frequently found in finance, population studies, and natural phenomena.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: Given the geometric sequence 4, 12, 36, ... Find the 7th term.
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Example 2: Calculate the sum of the first five terms of the sequence 3, 9, 27, ...
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Example 3: Determine if the sequence 8, 24, 72, ... is geometric and find the common ratio.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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If you want to find a term, just multiply and take your turn, r is constant, that’s your cue, to find the next from what you knew.
📖 Fascinating Stories
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In a land where trees grow beautifully, a garden grew where each tree tripled its height each year, showing the wonders of geometric growth.
🧠 Other Memory Gems
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MATH: Multiply, Add, Then Hunt for the next term! (For finding subsequent terms in geometric sequences.)
🎯 Super Acronyms
SAG
- Series
- Absolute ratio
- Growth—remembers the essence of geometric sequences.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Geometric Sequence
Definition:
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.
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Term: Common Ratio (r)
Definition:
A constant value that each term of a geometric sequence is multiplied by to get the next term.
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Term: nth term
Definition:
The term located at position n in a sequence, found using specific formulas.
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Term: Finite Geometric Series
Definition:
The sum of a fixed number of terms of a geometric sequence.
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Term: Infinite Geometric Series
Definition:
The sum of an infinite number of terms of a geometric sequence, which only converges if the common ratio is less than 1 in absolute value.