If 1, x, y, z, 16 are in geometric progression, then what is the value of x+y+z ?

Options

A. 8
B. 12
C. 14
D. 16

Explanation

To find the value of \( x + y + z \) given that \( 1, x, y, z, 16 \) are in geometric progression, we can follow these steps:

### Step 1: Understand the properties of a geometric progression (GP)
In a geometric progression, the ratio of consecutive terms is constant. If we denote the first term as \( a \) and the common ratio as \( r \), the terms can be expressed as:
- First term: \( a = 1 \)
- Second term: \( x = ar \)
- Third term: \( y = ar^2 \)
- Fourth term: \( z = ar^3 \)
- Fifth term: \( 16 = ar^4 \)

### Step 2: Set up the equations
From the above definitions, we can write:
1. \( x = r \)
2. \( y = r^2 \)
3. \( z = r^3 \)
4. \( 16 = r^4 \)

### Step 3: Solve for \( r \)
From the equation \( 16 = r^4 \), we can solve for \( r \):
\[ r = 16^{1/4} = 2 \]

### Step 4: Calculate \( x, y, z \)
Now that we have \( r \), we can find the values of \( x, y, z \):
- \( x = r = 2 \)
- \( y = r^2 = 2^2 = 4 \)
- \( z = r^3 = 2^3 = 8 \)

### Step 5: Find \( x + y + z \)
Now, we can sum these values:
\[ x + y + z = 2 + 4 + 8 = 14 \]

### Final Answer
Thus, the value of \( x + y + z \) is \( \boxed{14} \).

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