🌟 Level 2 - Topic 8: Introduction to Sequences and Series (Part 2) - Geometric Sequences & Series 🚀

1) Introduction to Geometric Sequences and Series 🚀

Welcome back to the exciting world of sequences and series! In Part 1, we explored arithmetic sequences and series, where each term is obtained by adding a constant difference. Now, get ready to discover a new type of sequence that grows (or shrinks!) in a different way – through multiplication! This is the realm of geometric sequences and series.

Imagine a sequence where each term is found by multiplying the previous term by a fixed number. This constant multiplier is called the common ratio. This kind of consistent multiplication creates a geometric sequence. Just like with arithmetic sequences, when you sum up the terms of a geometric sequence, you get a geometric series.

Geometric sequences and series are incredibly powerful tools in mathematics and have wide-ranging applications, from calculating compound interest in finance to understanding exponential growth and decay in science. They even appear in art and nature, describing patterns of scaling and proportion!

In Part 2, we will set our sights on:

• Understanding the core concept of geometric sequences and how they differ from arithmetic sequences.
• Learning how to find any term in a geometric sequence using a simple formula.
• Exploring geometric series and discovering how to calculate the sum of both finite and, in some cases, infinite geometric series.

Ready to multiply your knowledge and explore the fascinating world of geometric patterns? Let's get started! 🚀

2) What are Geometric Sequences? 🧮

2.1 Defining a Geometric Sequence

A geometric sequence is a list 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. This common ratio is the key to the pattern in a geometric sequence.

Let's denote the first term as \(a_1\) and the common ratio as \(r\). Then, a geometric sequence looks like this:

\( \{a_1, \underbrace{a_1 \cdot r}_{a_2}, \underbrace{a_1 \cdot r \cdot r}_{a_3}, \underbrace{a_1 \cdot r \cdot r \cdot r}_{a_4}, \ldots \} \)

Which simplifies to:

\( \{a_1, a_1r, a_1r^2, a_1r^3, \ldots \} \)

In essence, to get from one term to the next in a geometric sequence, you consistently multiply by the common ratio \(r\).

2.2 Examples of Geometric Sequences

Let's see some examples to clarify the concept:

• Example 1: Start with 2, and multiply by 3 each time: \( \{2, 6, 18, 54, 162, \ldots \} \). Here, the first term \(a_1 = 2\) and the common ratio \(r = 3\).
• Example 2: Start with 100, and multiply by 1/2 each time (halving each term): \( \{100, 50, 25, 12.5, 6.25, \ldots \} \). In this sequence, \(a_1 = 100\) and the common ratio \(r = \frac{1}{2}\).
• Example 3: Start with 5, and multiply by -2 each time (alternating signs): \( \{5, -10, 20, -40, 80, \ldots \} \). Here, \(a_1 = 5\) and the common ratio \(r = -2\). Notice how the signs alternate because the common ratio is negative.
• Example 4: A finite geometric sequence: \( \{3, 6, 12, 24, 48, 96 \} \). This sequence starts with \(a_1 = 3\), has a common ratio \(r = 2\), and contains exactly 6 terms.

2.3 Formula to Find Any Term: The n-th Term of a Geometric Sequence

Just like with arithmetic sequences, we have a formula to directly calculate any term in a geometric sequence without listing out all the preceding terms. To find the n-th term (\(a_n\)) of a geometric sequence, given the first term (\(a_1\)) and the common ratio (\(r\)), we use the formula:

\( a_n = a_1 \cdot r^{(n-1)} \)

This formula is incredibly useful for quickly finding terms far down the sequence. Notice that the common ratio \(r\) is raised to the power of \(n-1\).

Example 5: Finding the 7th Term Quickly

Let's take the geometric sequence: \( \{2, 6, 18, 54, 162, \ldots \} \). Use the formula to find the 7th term (\(a_7\)).

**Solution:**

• First term, \(a_1 = 2\).
• Common ratio, \(r = 3\).
• We want to find the 7th term, so \(n = 7\).

Using the formula: \( a_n = a_1 \cdot r^{(n-1)} \)

\( a_7 = 2 \cdot 3^{(7-1)} \) \( a_7 = 2 \cdot 3^6 \) \( a_7 = 2 \cdot 729 \) \( a_7 = 1458 \)

Thus, the 7th term of the sequence is 1458. Imagine how long it would take to list out all 7 terms manually! The formula makes it incredibly efficient.

3) What are Geometric Series? ➕

3.1 Defining a Geometric Series

As you might guess, a geometric series is what we get when we sum up the terms of a geometric sequence. If we have a geometric sequence \( \{a_1, a_1r, a_1r^2, a_1r^3, \ldots \} \), the corresponding geometric series is:

\( S = a_1 + a_1r + a_1r^2 + a_1r^3 + \ldots \)

Like arithmetic series, geometric series can also be finite (sum of a limited number of terms) or infinite (sum of infinitely many terms). However, the behavior of infinite geometric series is quite unique and depends crucially on the common ratio \(r\).

3.2 Formula for the Sum of a Finite Geometric Series

Let's first focus on how to calculate the sum of the first \(n\) terms of a geometric series, denoted as \(S_n\). There's a neat formula for this:

\( S_n = \frac{a_1(1 - r^n)}{1 - r} \), where \(r \neq 1\)

It's important to note that this formula works when the common ratio \(r\) is not equal to 1. If \(r = 1\), the geometric sequence becomes a simple arithmetic sequence with a common difference of 0, and the sum formula simplifies greatly (which we won't cover specifically as "geometric" for \(r=1\)).

Let's break down the formula:

• \(S_n\) – The sum of the first \(n\) terms.
• \(n\) – The number of terms being summed.
• \(a_1\) – The first term of the geometric sequence.
• \(r\) – The common ratio.

Example 6: Summing the First 6 Terms

Consider the geometric sequence \( \{3, 6, 12, 24, 48, 96 \} \). Let's find the sum of these 6 terms using the formula.

**Solution:**

• First term, \(a_1 = 3\).
• Common ratio, \(r = 2\) (since \(6/3 = 2\), \(12/6 = 2\), etc.).
• Number of terms to sum, \(n = 6\).

Using the formula: \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)

\( S_6 = \frac{3(1 - 2^6)}{1 - 2} \) \( S_6 = \frac{3(1 - 64)}{-1} \) \( S_6 = \frac{3(-63)}{-1} \) \( S_6 = 3 \times 63 \) \( S_6 = 189 \)

So, the sum of the geometric series \( 3 + 6 + 12 + 24 + 48 + 96 \) is 189. You can verify this by manually adding the terms!

3.3 Introduction to Infinite Geometric Series (and When They Make Sense!)

Now, let's think about infinite geometric series – what happens if we keep adding terms of a geometric sequence forever? It might seem like the sum would always become infinitely large, but surprisingly, for some geometric series, the sum approaches a finite value!

This intriguing behavior depends entirely on the common ratio \(r\). An infinite geometric series will have a finite sum only if the absolute value of the common ratio, \(|r|\), is less than 1 (i.e., \( -1 < r < 1 \)). In this case, we say the series converges. If \(|r| \geq 1\), the series diverges, meaning the sum grows without bound and does not approach a finite number.

Think about Example 2 from geometric sequences: \( \{100, 50, 25, 12.5, 6.25, \ldots \} \) with \(r = \frac{1}{2}\). As you keep going, the terms become smaller and smaller, approaching zero. It's plausible that if you add them all up, you might get a finite total.

On the other hand, in Example 1, \( \{2, 6, 18, 54, 162, \ldots \} \) with \(r = 3\), the terms are getting larger and larger. If you keep adding them, the sum will definitely grow to infinity!

3.4 Formula for the Sum of an Infinite Geometric Series (when it Converges)

When an infinite geometric series converges (i.e., when \(|r| < 1\)), we can calculate its sum using a remarkably simple formula:

\( S_\infty = \frac{a_1}{1 - r} \), for \(|r| < 1\)

This formula gives us the value that the sum of the infinite series approaches. It's a beautiful result showing that even sums of infinitely many numbers can be finite!

Example 7: Summing an Infinite Geometric Series

Let's find the sum of the infinite geometric series starting with 100 and having a common ratio of \(r = \frac{1}{2}\): \( 100 + 50 + 25 + 12.5 + 6.25 + \ldots \)

**Solution:**

• First term, \(a_1 = 100\).
• Common ratio, \(r = \frac{1}{2}\).
• Since \(|r| = |\frac{1}{2}| = \frac{1}{2} < 1\), the series converges, and we can use the formula.

Using the formula: \( S_\infty = \frac{a_1}{1 - r} \)

\( S_\infty = \frac{100}{1 - \frac{1}{2}} \) \( S_\infty = \frac{100}{\frac{1}{2}} \) \( S_\infty = 100 \times 2 \) \( S_\infty = 200 \)

Therefore, the sum of the infinite geometric series \( 100 + 50 + 25 + 12.5 + 6.25 + \ldots \) approaches 200. This means if you kept adding terms of this sequence forever, the total would get closer and closer to 200, without ever exceeding it!

4) Examples (Detailed) 🍀

Example 8: Working with Geometric Sequences and Series

Let's explore a geometric sequence with a first term \(a_1 = 16\) and a common ratio \(r = -\frac{1}{2}\). Let's tackle a few tasks:

1. Challenge 1: Write down the first 4 terms of this geometric sequence.
2. Challenge 2: Find the 10th term (\(a_{10}\)).
3. Challenge 3: Calculate the sum of the first 8 terms (\(S_8\)).
4. Challenge 4: Can we find the sum of this geometric series if it goes on infinitely? If yes, calculate it.

**Let's solve these challenges step-by-step:**

1. Solution to Challenge 1: First 4 terms

   Starting with \(a_1 = 16\) and multiplying by \(r = -\frac{1}{2}\) for each subsequent term:

   \(a_1 = 16\)
   \(a_2 = a_1 \cdot r = 16 \times (-\frac{1}{2}) = -8\)
   \(a_3 = a_2 \cdot r = -8 \times (-\frac{1}{2}) = 4\)
   \(a_4 = a_3 \cdot r = 4 \times (-\frac{1}{2}) = -2\)
   

   The first 4 terms are: \( \{16, -8, 4, -2\} \). Notice the alternating signs due to the negative common ratio.

2. Solution to Challenge 2: The 10th Term (\(a_{10}\))

   Using the n-th term formula: \( a_n = a_1 \cdot r^{(n-1)} \) with \(n = 10\), \(a_1 = 16\), and \(r = -\frac{1}{2}\):

   \( a_{10} = 16 \cdot (-\frac{1}{2})^{(10-1)} \)
   \( a_{10} = 16 \cdot (-\frac{1}{2})^{9} \)
   \( a_{10} = 16 \cdot (-\frac{1}{512}) \)
   \( a_{10} = -\frac{16}{512} = -\frac{1}{32} \)

   The 10th term is \(-\frac{1}{32}\).

3. Solution to Challenge 3: Sum of the First 8 Terms (\(S_8\))

   For the sum of the first 8 terms, we use the formula \( S_n = \frac{a_1(1 - r^n)}{1 - r} \) with \(n = 8\), \(a_1 = 16\), and \(r = -\frac{1}{2}\):

   \( S_8 = \frac{16(1 - (-\frac{1}{2})^8)}{1 - (-\frac{1}{2})} \)
   \( S_8 = \frac{16(1 - \frac{1}{256})}{1 + \frac{1}{2}} \)
   \( S_8 = \frac{16(1 - \frac{1}{256})}{\frac{3}{2}} \)
   \( S_8 = \frac{16 \times \frac{255}{256}}{\frac{3}{2}} \)
   \( S_8 = \frac{16 \times 255}{256} \times \frac{2}{3} \)
   \( S_8 = \frac{2 \times 255}{24} = \frac{255}{12} = \frac{85}{4} = 21.25 \)

   The sum of the first 8 terms is 21.25.

4. Solution to Challenge 4: Sum of the Infinite Series?

   Yes, we can find the sum of the infinite geometric series because the common ratio \(r = -\frac{1}{2}\) has an absolute value \(|r| = |-\frac{1}{2}| = \frac{1}{2} < 1\), so the series converges.

   Using the formula for infinite geometric series: \( S_\infty = \frac{a_1}{1 - r} \) with \(a_1 = 16\) and \(r = -\frac{1}{2}\):

   \( S_\infty = \frac{16}{1 - (-\frac{1}{2})} \)
   \( S_\infty = \frac{16}{1 + \frac{1}{2}} \)
   \( S_\infty = \frac{16}{\frac{3}{2}} \)
   \( S_\infty = 16 \times \frac{2}{3} \)
   \( S_\infty = \frac{32}{3} = 10\frac{2}{3} \approx 10.67 \)

   The sum of this infinite geometric series is \( \frac{32}{3} \) or approximately 10.67.

5) Practice Questions 🎯

5.1 Fundamental – Build Skills

5.2 Challenging – Push Limits 💪🚀

6) Summary 🎉

• Geometric sequences are ordered lists where each term is found by multiplying the previous term by a constant common ratio (\(r\)).
• The n-th term of a geometric sequence is given by \( a_n = a_1 \cdot r^{(n-1)} \).
• A geometric series is the sum of the terms of a geometric sequence.
• The sum of the first \(n\) terms of a finite geometric series is \( S_n = \frac{a_1(1 - r^n)}{1 - r} \) (for \(r \neq 1\)).
• An infinite geometric series converges to a finite sum only if \(|r| < 1\).
• The sum of a convergent infinite geometric series is \( S_\infty = \frac{a_1}{1 - r} \).

Congratulations! You've now completed your exploration of both arithmetic and geometric sequences and series! You've learned to identify these sequences, find any term, and calculate their sums – both finite and infinite in the geometric case. These skills are fundamental in mathematics and have applications in many areas. Keep practicing, and you'll become proficient in recognizing and working with these powerful mathematical patterns! 🌟 What's next? Stay tuned for more exciting topics in Level 2 as we continue to build your CodeMathFusion skills!