$$ 20(k-4) ;(k+1) ; m ; 5 k $$ is a set of numbers. The first three terms form an arithmetic sequence and the last three form a geometric sequence. Determine the values of $$ k $$ and $$ m $$, if $$ k, m0 $$.

Question

$$ 20(k-4) ;(k+1) ; m ; 5 k $$ is a set of numbers. The first three terms form an arithmetic sequence and the last three form a geometric sequence. Determine the values of $$ k $$ and $$ m $$, if $$ k, m>0 $$.

UpStudy AI · Accepted Answer

To determine the values of $$ k $$ and $$ m $$ given the set of numbers $$ \{20(k-4),\ (k+1),\ m,\ 5k\} $$ with the first three forming an arithmetic sequence and the last three forming a geometric sequence, follow these steps:

### 1. Arithmetic Sequence Condition
For the first three terms to form an arithmetic sequence, the difference between consecutive terms must be constant. Therefore:

$$
(k + 1) - 20(k - 4) = m - (k + 1)
$$

Simplifying:

$$
k + 1 - 20k + 80 = m - k - 1 \
-19k + 81 = m - k - 1 \
-18k + 82 = m
$$

Thus, we have:
$$
m = -18k + 82 \quad \text{(Equation 1)}
$$

### 2. Geometric Sequence Condition
For the last three terms to form a geometric sequence, the ratio between consecutive terms must be constant. Therefore:

$$
\frac{m}{k + 1} = \frac{5k}{m}
$$

Cross-multiplying gives:

$$
m^2 = 5k(k + 1)
$$

Substituting $$ m $$ from Equation 1:

$$
(-18k + 82)^2 = 5k(k + 1)
$$

Expanding the left side:

$$
324k^2 - 2952k + 6724 = 5k^2 + 5k
$$

Bringing all terms to one side:

$$
319k^2 - 2957k + 6724 = 0
$$

### 3. Solving the Quadratic Equation
Using the quadratic formula:

$$
k = \frac{2957 \pm \sqrt{2957^2 - 4 \cdot 319 \cdot 6724}}{2 \cdot 319}
$$

Calculating the discriminant:

$$
\Delta = 2957^2 - 4 \cdot 319 \cdot 6724 = 164,025
$$

Since $$ \sqrt{164,025} = 405 $$, the solutions are:

$$
k = \frac{2957 \pm 405}{638}
$$

This yields two potential solutions:

1. $$ k = \frac{2957 + 405}{638} = \frac{3362}{638} = \frac{1681}{319} $$ (Not a positive integer)
2. $$ k = \frac{2957 - 405}{638} = \frac{2552}{638} = 4 $$

### 4. Validating the Solution
Substituting $$ k = 4 $$ into Equation 1:

$$
m = -18(4) + 82 = -72 + 82 = 10
$$

Both $$ k = 4 $$ and $$ m = 10 $$ are positive, satisfying the given conditions.

### **Final Answer:**
$$
k = 4 \quad \text{and} \quad m = 10
$$
$$ k = 4 $$ and $$ m = 10 $$.

is a set of numbers. The first three terms form
an arithmetic sequence and the last three form a geometric sequence.
Determine the values of and , if .

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is a set of numbers. The first three terms form an arithmetic sequence and the last three form a geometric sequence. Determine the values of and , if .