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
2nd differences
2nd differences
1st differences
1st differences
The level differences of this polynomial that will result in a constant value is the 1st differences.
To determine the level differences of a polynomial, we need to find the differences between consecutive terms of the polynomial. In this case, the polynomial is given by the formula n(n+1)/2, which represents the sum of a sequence of consecutive numbers from 1 to n.
Let's calculate the 1st differences first:
To find the 1st differences, we subtract consecutive terms of the polynomial. For example, if we calculate the differences between terms n=1 and n=2, we get:
(2(2+1)/2) - (1(1+1)/2) = 1 - 1 = 0.
Similarly, for the differences between terms n=2 and n=3:
(3(3+1)/2) - (2(2+1)/2) = 3 - 1 = 2.
If we continue this process, we will find that the 1st differences form a sequence of consecutive numbers:
1st differences: 0, 2, 3, 4, ...
Now let's calculate the 2nd differences:
To find the 2nd differences, we subtract consecutive terms of the 1st differences. Continuing from the 1st differences:
2nd differences: 2 - 0 = 2, 3 - 2 = 1, 4 - 3 = 1, ...
If we continue this process, we will find that the 2nd differences form a constant value, which is:
2nd differences: 2, 1, 1, 1, ...
Therefore, the level differences of this polynomial that result in a constant value are the 2nd differences.