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

1st differences
1st differences

2nd differences
2nd differences

3rd differences
3rd differences

4th differences

Answer 1

To determine which level differences of the polynomial \( S_n = \frac{n(n+1)}{2} \) is a constant, we can compute the differences as follows:

1. **Find the first differences**:
\( S_n = \frac{n(n+1)}{2} \), so:
\[
S_1 = \frac{1(1+1)}{2} = 1
\]
\[
S_2 = \frac{2(2+1)}{2} = 3
\]
\[
S_3 = \frac{3(3+1)}{2} = 6
\]
\[
S_4 = \frac{4(4+1)}{2} = 10
\]
\[
S_5 = \frac{5(5+1)}{2} = 15
\]

The first differences are:
\[
S_2 - S_1 = 3 - 1 = 2
\]
\[
S_3 - S_2 = 6 - 3 = 3
\]
\[
S_4 - S_3 = 10 - 6 = 4
\]
\[
S_5 - S_4 = 15 - 10 = 5
\]

First differences: \( 2, 3, 4, 5 \)

2. **Find the second differences**:
\[
3 - 2 = 1
\]
\[
4 - 3 = 1
\]
\[
5 - 4 = 1
\]

Second differences: \( 1, 1, 1 \)

Since the second differences are constant, we conclude:

**The answer is: 2nd differences.**