$$ x ; y ; 8 ; \ldots $$ is a geometric sequence and $$ 1 ; x+1 ; y+1 ; \ldots $$ is an arithmetic sequence. Determine the values of $$ x $$ and $$ y $$.

Question

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Given:
1. **Geometric Sequence**: $$ x ; y ; 8 ; \ldots $$
2. **Arithmetic Sequence**: $$ 1 ; x+1 ; y+1 ; \ldots $$

### Step 1: Geometric Sequence

For a sequence to be geometric, the ratio between consecutive terms must be constant. Let the common ratio be $$ r $$.

- $$ y = r \cdot x $$
- $$ 8 = r \cdot y $$

Substituting $$ y $$ from the first equation into the second:
$$ 8 = r \cdot (r \cdot x) = r^2 \cdot x $$
$$ r^2 \cdot x = 8 \quad \text{(Equation 1)} $$

### Step 2: Arithmetic Sequence

For a sequence to be arithmetic, the difference between consecutive terms must be constant. Let the common difference be $$ d $$.

- $$ (x + 1) - 1 = x $$
- $$ (y + 1) - (x + 1) = y - x $$

Setting the differences equal:
$$ x = y - x $$
$$ y = 2x \quad \text{(Equation 2)} $$

### Step 3: Solving the Equations

From Equation 2: $$ y = 2x $$

Substitute $$ y $$ into Equation 1:
$$ r^2 \cdot x = 8 $$

From $$ y = r \cdot x $$ and $$ y = 2x $$:
$$ r \cdot x = 2x $$
Assuming $$ x \neq 0 $$:
$$ r = 2 $$

Now substitute $$ r = 2 $$ back into Equation 1:
$$ (2)^2 \cdot x = 8 $$
$$ 4x = 8 $$
$$ x = 2 $$

Using Equation 2:
$$ y = 2 \cdot 2 = 4 $$

### Verification:

- **Geometric Sequence**: $$ 2 ; 4 ; 8 ; \ldots $$ with ratio $$ r = 2 $$
- **Arithmetic Sequence**: $$ 1 ; 3 ; 5 ; \ldots $$ with difference $$ d = 2 $$

Both sequences satisfy the given conditions.

### **Answer:**
$$ x = 2 $$ and $$ y = 4 $$
$$ x = 2 $$ and $$ y = 4 $$.

is a geometric sequence and is an arithmetic sequence.
Determine the values of and .

Upstudy AI Solution

Answer

Solution

Given:

Step 1: Geometric Sequence

Step 2: Arithmetic Sequence

Step 3: Solving the Equations

Verification:

Answer:

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