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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

2nd differences
2nd differences

4th differences
4th differences

3rd differences
3rd differences

1st differences

1 answer

The sum of the first \( n \) natural numbers is given by the formula \(\frac{n(n+1)}{2}\), which is a quadratic polynomial of the form \( \frac{1}{2}n^2 + \frac{1}{2}n \).

To determine which level of differences is constant, we analyze the polynomial:

  1. 1st differences: This involves calculating the differences of consecutive terms of the sequence (i.e., \( S(n) - S(n-1) \)).
  2. 2nd differences: This involves calculating the differences of the 1st differences.
  3. 3rd differences: This involves calculating the differences of the 2nd differences.
  4. 4th differences: This involves calculating the differences of the 3rd differences.

Since the sequence is described by a quadratic polynomial, the 2nd differences will be constant. This is a general property of polynomials: if the polynomial is of degree \( k \), then the \( (k+1) \)-th differences are constant (and in this case, \( k = 2 \) for a quadratic polynomial).

Thus, the correct response is:

2nd differences