If `3^(k+1)-3^(k)=m`, what is `3^(k+2)` in terms of m?

Options

A. `(4)/(3)m`
B. `3m`
C. `(9)/(2)m`
D. `9m`

Answer

To solve the equation \(3^{k+1} - 3^k = m\) and find \(3^{k+2}\) in terms of \(m\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3^{k+1} - 3^k = m \]

### Step 2: Factor out \(3^k\)
Notice that \(3^{k+1}\) can be expressed as \(3^1 \cdot 3^k\). Thus, we can rewrite the left side:
\[ 3^1 \cdot 3^k - 3^k = m \]
This simplifies to:
\[ 3 \cdot 3^k - 3^k = m \]

### Step 3: Factor out \(3^k\)
Now, we can factor out \(3^k\) from the left-hand side:
\[ (3 - 1) \cdot 3^k = m \]
This simplifies to:
\[ 2 \cdot 3^k = m \]

### Step 4: Solve for \(3^k\)
Now, we can solve for \(3^k\):
\[ 3^k = \frac{m}{2} \]

### Step 5: Find \(3^{k+2}\)
Next, we want to find \(3^{k+2}\). We can express \(3^{k+2}\) as:
\[ 3^{k+2} = 3^2 \cdot 3^k \]
Substituting \(3^k\) from the previous step:
\[ 3^{k+2} = 9 \cdot \frac{m}{2} \]

### Step 6: Simplify the expression
Now, we simplify this expression:
\[ 3^{k+2} = \frac{9m}{2} \]

### Final Answer
Thus, the value of \(3^{k+2}\) in terms of \(m\) is:
\[ 3^{k+2} = \frac{9m}{2} \]

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