The sum of a sequence of consecutive numbers from 1 to n is given by the formula n(n+1)2

Question

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.