4 1.3.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{,} $$as $$ n \rightarrow \infty $$ (2) 1.4 CASE 3: $$ r=1 $$ 1.4.1 What is the sum of the geometric series $$ S_{-}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=1 ? $$ 1.4.2 Make a conjecture with regard to $$ r^{\prime \prime} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$ $$ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=-1 ? $$ i.e. is the sequence divergent or convergent? 1.6.1 Write THREE possible values of $$ r $$ such that $$ -1

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### 1.3.2 Conjecture Regarding $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$

In a geometric series, the sum $$ S_n $$ is given by:

$$
S_n = a + ar + ar^2 + \ldots + ar^{n-1}
$$

The formula for the sum of the first $$ n $$ terms of a geometric series is:

$$
S_n = a \frac{1 - r^n}{1 - r} \quad \text{if } r \neq 1
$$

As $$ n \rightarrow \infty $$:

- If $$ |r| < 1 $$, then $$ r^n \rightarrow 0 $$, and thus $$ S_n \rightarrow \frac{a}{1 - r} $$.
- If $$ |r| > 1 $$, then $$ r^n \rightarrow \infty $$, and thus $$ S_n $$ diverges.
- If $$ r = 1 $$, then $$ S_n = na $$, which also diverges as $$ n \rightarrow \infty $$.

**Conjecture**: As $$ n \rightarrow \infty $$:
- If $$ |r| < 1 $$, $$ S_n $$ converges to $$ \frac{a}{1 - r} $$.
- If $$ |r| \geq 1 $$, $$ S_n $$ diverges.

### 1.4 CASE 3: $$ r=1 $$

#### 1.4.1 Sum of the Geometric Series when $$ r=1 $$

When $$ r = 1 $$:

$$
S_n = a + a + a + \ldots + a \quad (n \text{ terms}) = na
$$

Thus, the sum of the geometric series is:

$$
S_n = na
$$

#### 1.4.2 Conjecture Regarding $$ r = -1 $$

When $$ r = -1 $$:

$$
S_n = a - a + a - a + \ldots
$$

The terms alternate between $$ a $$ and $$ -a $$. The sum depends on whether $$ n $$ is even or odd:

- If $$ n $$ is even, $$ S_n = 0 $$.
- If $$ n $$ is odd, $$ S_n = a $$.

**Conjecture**: The sequence is divergent because it does not approach a single value as $$ n \rightarrow \infty $$.

### 1.6.1 Possible Values of $$ r $$ such that $$ -1 < r < 1 $$

Three possible values of $$ r $$ in the interval $$ -1 < r < 1 $$ could be:

1. $$ r = 0.5 $$
2. $$ r = -0.5 $$
3. $$ r = 0.9 $$

### 1.6.5 and 1.6.6 Areas of Paper Pieces

1. **Step 4**: The first piece of paper has an area of $$ \frac{1}{2} $$ the area of A4. The area of A4 is approximately $$ 0.297 \, m^2 $$ or $$ 297 \, cm^2 $$. Thus, the area of the first piece is:

$$
\text{Area}_1 = \frac{1}{2} \times 297 \, cm^2 = 148.5 \, cm^2
$$

2. **Step 5**: The second piece of paper has an area of $$ \frac{1}{4} $$ the area of A4:

$$
\text{Area}_2 = \frac{1}{4} \times 297 \, cm^2 = 74.25 \, cm^2
$$

3. **Next Three Areas**: Continuing this pattern, the areas of the next three pieces of paper would be:

- Third piece: $$ \frac{1}{8} \times 297 \, cm^2 = 37.3125 \, cm^2 $$
- Fourth piece: $$ \frac{1}{16} \times 297 \, cm^2 = 18.65625 \, cm^2 $$
- Fifth piece: $$ \frac{1}{32} \times 297 \, cm^2 = 9.328125 \, cm^2 $$

Thus, the areas of the next three pieces of paper are approximately:

1. $$ 37.31 \, cm^2 $$
2. $$ 18.66 \, cm^2 $$
3. $$ 9.33 \, cm^2 $$
**Summary:** 1. **Conjecture for $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$:** - If $$ |r| < 1 $$, the sum $$ S_n $$ converges to $$ \frac{a}{1 - r} $$. - If $$ |r| \geq 1 $$, the sum $$ S_n $$ diverges. 2. **Sum of Geometric Series when $$ r=1 $$:** - $$ S_n = na $$ 3. **Conjecture for $$ r = -1 $$:** - The sequence is divergent because it alternates and does not approach a single value. 4. **Possible Values of $$ r $$ ($$ -1 < r < 1 $$):** - 0.5, -0.5, 0.9 5. **Areas of Paper Pieces:** - First piece: 148.5 cm² - Second piece: 74.25 cm² - Next three pieces: 37.31 cm², 18.66 cm², 9.33 cm²

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions