$$ x-2: \frac{1 x-2 y}{4} $$ and $$ \frac{1 x-211}{16} $$ are the first three terms of a geometric sequenc 2.1 Dor which valuers) of $$ x $$ is the sequence convergent? 2.2 Determine the sum to infinity if $$ x-1 $$. Conslder the geometric series: $$ 45+15-5+\ldots $$ 3.1 Does the sequence converge or diverge? Justify your answer. 3.2 Determine the sum to infinity of the series, if it exists.

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Part 2: Geometric Sequence

We have the first three terms of a geometric sequence given as:
1. $$ a_1 = x - 2 $$
2. $$ a_2 = \frac{1(x - 2y)}{4} $$
3. $$ a_3 = \frac{1(x - 211)}{16} $$

#### 2.1 For which values of $$ x $$ is the sequence convergent?

For a geometric sequence to be convergent, the common ratio $$ r $$ must satisfy $$ |r| < 1 $$.

The common ratio $$ r $$ can be calculated as:
$$
r = \frac{a_2}{a_1} = \frac{\frac{1(x - 2y)}{4}}{x - 2}
$$
and
$$
r = \frac{a_3}{a_2} = \frac{\frac{1(x - 211)}{16}}{\frac{1(x - 2y)}{4}}
$$

We will set these two expressions for $$ r $$ equal to each other and solve for $$ x $$.

#### 2.2 Determine the sum to infinity if $$ x - 1 $$.

The sum to infinity $$ S $$ of a geometric series is given by:
$$
S = \frac{a_1}{1 - r}
$$
where $$ a_1 $$ is the first term and $$ r $$ is the common ratio.

Now, let's calculate the values for $$ r $$ and $$ S $$.

### Part 3: Geometric Series

We have the series:
$$
45 + 15 - 5 + \ldots
$$

#### 3.1 Does the sequence converge or diverge? Justify your answer.

To determine convergence, we need to find the common ratio $$ r $$:
$$
r = \frac{15}{45} = \frac{1}{3}
$$
and
$$
r = \frac{-5}{15} = -\frac{1}{3}
$$

Since $$ |r| = \frac{1}{3} < 1 $$, the series converges.

#### 3.2 Determine the sum to infinity of the series, if it exists.

The sum to infinity $$ S $$ is given by:
$$
S = \frac{a_1}{1 - r}
$$
where $$ a_1 = 45 $$ and $$ r = \frac{1}{3} $$.

Now, let's perform the calculations for both parts.

### Calculations

1. **For Part 2:**
   - Calculate $$ r $$ and find the values of $$ x $$ for convergence.
   - Calculate the sum to infinity.

2. **For Part 3:**
   - Confirm convergence and calculate the sum to infinity.

Let's proceed with the calculations.

#### Part 2 Calculations

1. Calculate $$ r $$ from $$ a_1 $$ and $$ a_2 $$:
   $$
   r = \frac{\frac{1(x - 2y)}{4}}{x - 2}
   $$
   and from $$ a_2 $$ and $$ a_3 $$:
   $$
   r = \frac{\frac{1(x - 211)}{16}}{\frac{1(x - 2y)}{4}}
   $$

2. Set the two expressions for $$ r $$ equal and solve for $$ x $$.

#### Part 3 Calculations

1. Confirm convergence:
   $$
   |r| = \frac{1}{3} < 1
   $$

2. Calculate the sum to infinity:
   $$
   S = \frac{45}{1 - \frac{1}{3}} = \frac{45}{\frac{2}{3}} = 45 \cdot \frac{3}{2} = 67.5
   $$

Now, let's perform the calculations for Part 2.

### Part 2 Calculations

We will solve the equations for $$ r $$ and find the values of $$ x $$.

Let's calculate the common ratio $$ r $$ and find the values of $$ x $$ for convergence.

#### Solving for $$ x $$