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

. Which level differences of this polynomial is a constant value?(1 point)
Responses

4th differences
4th differences

3rd differences
3rd differences

1st differences
1st differences

2nd differences

1 answer

The sum of the first \( n \) consecutive numbers is given by the formula:

\[ S(n) = \frac{n(n+1)}{2} \]

This can be expressed as a polynomial function of degree 2 (specifically, a quadratic polynomial). To find which level of differences is constant, we need to consider the properties of polynomials:

  • For a polynomial of degree \( k \), the \( k \)-th differences are constant.
  • The first differences of a quadratic polynomial will be linear (degree 1), hence the second differences will be constant.

Since the given expression is a quadratic polynomial (degree 2), the second differences will be constant.

The correct response is therefore:

2nd differences