Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take mock test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Geometric Sequences
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we'll discuss geometric sequences, which are sequences where each term is found by multiplying the previous term by a constant. Can anyone tell me what that constant is called?
Is it the common ratio?
Correct! The common ratio is represented by 'r'. For example, if we start with 2 and our common ratio is 3, the sequence would be 2, 6, 18, 54, and so on. Can anyone think of a situation where geometric sequences might be used?
Maybe in population growth?
Exactly! In population studies, if a population triples every year, that's a geometric sequence.
Finding the n-th Term
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now that we've covered what a geometric sequence is, let's look at how to find the n-th term. The formula is T = ar^(n-1). Who can tell me what 'n' represents?
'n' is the position of the term in the sequence.
Great! Let’s calculate the 5th term of the sequence where a = 3 and r = 2. Can someone apply the formula?
T = 3 * 2^(5-1) = 3 * 16 = 48!
Well done! And just like that, you can find any term in a geometric sequence.
Sum of Finite Geometric Series
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Next, let’s learn about the sum of the first n terms of a geometric sequence. The formula is S = a(1 - r^n) / (1 - r). Can someone explain when we would use this formula?
When we want to know the total of a certain number of terms in a geometric sequence?
Exactly right! Let me give you an example: what is the sum of the first 5 terms of the sequence where a = 2 and r = 3?
Using the formula: S = 2(1 - 3^5) / (1 - 3) which simplifies to 242.
Excellent job!
Infinite Geometric Series
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now let's look at infinite geometric series. We only find a finite sum if the absolute value of our common ratio is less than one. Can anyone provide me the formula for the sum of an infinite geometric series?
S∞ = a / (1 - r) when |r| < 1!
Yes! Let’s do an example. What is the sum of the infinite series: 5 + 2.5 + 1.25 + ...? What’s 'a' and 'r' in this case?
Here, a is 5 and r is 0.5. The sum would be S∞ = 5 / (1 - 0.5) = 10.
Well done! You've grasped a significant concept!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
Geometric sequences involve terms in which each is derived from multiplying the previous term by a common ratio. Understanding geometric sequences allows for the exploration of various applications such as compound interest and exponential growth. This section provides a comprehensive overview of formulas for the n-th term, sums of finite and infinite geometric series, and how to apply these concepts in real-life scenarios.
Detailed
In this chapter summary of geometric sequences, we explore the foundational concepts of this type of sequence where each term results from multiplying the preceding term by a constant known as the common ratio (r). The section delves into important formulas: the general term formula (T = ar^(n-1)), the sum of a finite series (S = a(1 - r^n) / (1 - r)), and the formula for an infinite series (S∞ = a / (1 - r), valid when |r| < 1). Skills in recognizing geometric sequences, finding terms, and solving real-world problems such as compound interest are emphasized. The significance of understanding these sequences extends to various fields including finance, biology, and engineering, showcasing their applicability in growth and decay scenarios.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
General Term (n-th term)
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The general term of a geometric sequence is expressed as:
📌 Formula: 𝑇 = 𝑎𝑟^{n−1}
Detailed Explanation
The general term (or n-th term) of a geometric sequence allows us to find any term in the sequence based on its position, n. In the formula, '𝑎' represents the first term of the sequence, and '𝑟' is the common ratio, which tells us how much we multiply the previous term to get the next term. For example, if the first term is 2 (𝑎 = 2) and the common ratio is 3 (𝑟 = 3), the formula will yield the terms of the sequence for any n, such as the 1st term (T = 23^0 = 2) or the 5th term (T = 23^4 = 162).
Examples & Analogies
Imagine you're stacking blocks. You start with one block (the first term), and with each hour, you decide to triple your stack (that's your common ratio). If you want to know how high your stack is after n hours, you can use this formula to find out just how many blocks you'll have!
Finite Series Sum
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The sum of the first n terms of a geometric sequence is given by:
📌 Formula (when 𝑟 ≠ 1): 𝑆 = 𝑎 ⋅ \frac{1 − 𝑟^{n}}{1−𝑟}
Detailed Explanation
This formula allows you to calculate the total of a specified number of terms in the sequence. Here, 'S' represents the sum of the first n terms, 'a' is still the first term, and 'r' is the common ratio. If the common ratio is not equal to 1 (this means the sequence is actually changing as terms progress), this formula is valid. For example, if you want the sum of the first 5 terms of a geometric sequence where the first term is 2 and the common ratio is 3, you would substitute these values into the formula to get your result.
Examples & Analogies
Think about earning monthly allowances that increase over time. If you receive an allowance that doubles each month (3 being your common ratio), you can use this formula to figure out how much you'll have received after a certain number of months!
Infinite Series Sum
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
An infinite geometric series can converge if the absolute value of the common ratio is less than 1:
📌 Formula: 𝑆 = \frac{𝑎}{1−𝑟}, for |𝑟| < 1
Detailed Explanation
An infinite geometric series has an unlimited number of terms. However, for the sum to remain finite (converge), the common ratio must be less than one in absolute value. This means each successive term becomes smaller and smaller. The formula gives a way to calculate the total sum of these terms. For instance, if the first term is 5 and the common ratio is 0.5, you can apply this formula to find that the total sum of the series equals 10, even though you keep adding more terms indefinitely.
Examples & Analogies
Imagine saving money in a bank where you earn interest, but that interest decreases over time because it's only a fraction of what you already have. If your initial amount is high, your total savings (in terms of interest) will keep growing, but not excessively high, because it's based on a decreasing pattern (the common ratio).
Common Ratio and Term Detection
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
To determine if a sequence is geometric:
1. Divide any term by the previous term:
𝑟 = \frac{𝑇_{n+1}}{𝑇_{n}}
2. Check if the ratio is constant.
Detailed Explanation
To verify if a sequence is geometric, you start by dividing one term of the sequence by the term that comes immediately before it. This will give you the common ratio. If you find that this ratio is constant across the entire sequence, then the sequence is indeed geometric. For example, if you have a sequence like 2, 6, 18, 54, if you divide each term by its predecessor, you'll consistently get the same ratio, confirming it's geometric.
Examples & Analogies
Picture a tree that grows where each branch splits off into three branches. If you look at any generation of branches and compare the number of branches each time, you can determine that the growth rule (the common ratio) remains consistent each season. That’s how you would figure out the growth pattern of the tree!
Glossary of Terms
Review the Definitions for terms.
-
Term: Geometric Sequence
Definition:
A sequence where each term is obtained by multiplying the previous term by a constant called the common ratio.
-
Term: Common Ratio (r)
Definition:
The constant value multiplied to obtain consecutive terms in a geometric sequence.
-
Term: nth Term
Definition:
The term at position 'n' in a geometric sequence, calculated using the formula T = ar^(n-1).
-
Term: Finite Series Sum
Definition:
The sum of a specified number of terms in a geometric sequence, calculated using the formula S = a(1 - r^n) / (1 - r).
-
Term: Infinite Series Sum
Definition:
The sum of an infinite geometric series, which exists under the condition |r| < 1, calculated using the formula S∞ = a / (1 - r).