Question

The sum of a sequence of consecutive numbers from 1 to n is given by the formula n(n+1)2
𝑛
(
𝑛
+
1
)
2
. Which level differences of this polynomial is a constant value?(1 point)
Responses

4th differences

3rd differences

1st differences

2nd differences

Answer 1

To determine which level of differences of the polynomial \( S(n) = \frac{n(n+1)}{2} \) is a constant value, let's analyze the polynomial.

### Step 1: Rewrite the formula
The formula for the sum of the first \( n \) natural numbers can be simplified to:
\[
S(n) = \frac{n^2 + n}{2} = \frac{1}{2}n^2 + \frac{1}{2}n
\]

### Step 2: Identify the degree
This polynomial is a quadratic polynomial because it has the highest power of \( n \) as 2 (i.e., \( \frac{1}{2}n^2 \)).

### Step 3: Find the differences
To analyze the differences:

1. **1st differences**: \( S(n+1) - S(n) \)
2. **2nd differences**: 1st differences of the 1st differences
3. **3rd differences**: 1st differences of the 2nd differences
4. **4th differences**: 1st differences of the 3rd differences

The first difference will be linear since the function is quadratic.

### Step 4: Constant differences
For a polynomial of degree \( d \):
- The \( d \)-th differences will be constant.
- Therefore, since \( S(n) \) is a quadratic polynomial (degree 2), the **2nd differences** will be constant.

### Conclusion
The answer is **2nd differences**.