If `u_(0) = 8 , u_(1) = 3 , u_(2) = 12 , u_(3) = 51` , then the value of `Delta^(3) u_(0)` is

Options

A. 12
B. 14
C. 16
D. 18

Explanation

To find the value of \( \Delta^3 u_0 \) given the values \( u_0 = 8 \), \( u_1 = 3 \), \( u_2 = 12 \), and \( u_3 = 51 \), we will use the formula for the finite difference operator.

### Step-by-Step Solution:

1. **Understanding the Finite Difference Operator**:
The \( n \)-th finite difference \( \Delta^n u(x) \) can be expressed in terms of the values of the function at discrete points. Specifically, \( \Delta^3 u_0 \) can be calculated using the formula:
\[ \Delta^3 u_0 = u_3 - 3u_2 + 3u_1 - u_0 \]

2. **Substituting the Values**:
Now, we substitute the known values into the formula:
- \( u_0 = 8 \)
- \( u_1 = 3 \)
- \( u_2 = 12 \)
- \( u_3 = 51 \)

Thus, we have:
\[ \Delta^3 u_0 = 51 - 3(12) + 3(3) - 8 \]

3. **Calculating Each Term**:
- First, calculate \( 3(12) = 36 \)
- Then, calculate \( 3(3) = 9 \)

Now substituting these values back into the equation:
\[ \Delta^3 u_0 = 51 - 36 + 9 - 8 \]

4. **Performing the Arithmetic**:
- Start with \( 51 - 36 = 15 \)
- Next, add \( 15 + 9 = 24 \)
- Finally, subtract \( 24 - 8 = 16 \)

5. **Final Result**:
Thus, the value of \( \Delta^3 u_0 \) is:
\[ \Delta^3 u_0 = 16 \]

### Conclusion:
The required value of \( \Delta^3 u_0 \) is \( 16 \).