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

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

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

This expression is a polynomial of degree 2. To determine the level of differences that is constant, we can analyze the differences of the sequence values generated by this polynomial.

1. First differences: Calculate \( S(n+1) - S(n) \) \[ S(n) = \frac{n(n+1)}{2} \quad \text{and} \quad S(n+1) = \frac{(n+1)(n+2)}{2} \] \[ S(n+1) - S(n) = \frac{(n+1)(n+2)}{2} - \frac{n(n+1)}{2} \]

   Simplifying, \[ = \frac{(n+1)(n+2 - n)}{2} = \frac{(n+1)(2)}{2} = n + 1 \]

   The first differences are \( n + 1 \), which is a linear function (degree 1).

2. Second differences: Calculate \( (S(n+1) - S(n)) - (S(n) - S(n-1)) \) The first differences \( n + 1 \) are linearly increasing, so: \[ (n + 1) - ((n-1) + 1) = n + 1 - n = 1 \] The second differences are constant and equal to 1.

Since the second differences are constant, we conclude that the answer is:

2nd differences.