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)
a. 4th differences
b. 3rd differences
c. 2nd differences
d. 1st differences
c. 2nd differences
The level differences of a polynomial refer to the differences between consecutive terms of the polynomial.
In this case, the polynomial is represented by the formula n(n+1)/2, which gives the sum of a sequence of consecutive numbers from 1 to n.
To find the level differences, we can calculate the differences between consecutive terms of the polynomial.
Let's calculate the differences for the first few terms of the polynomial:
For n = 1: 1(1+1)/2 = 1
For n = 2: 2(2+1)/2 = 3
For n = 3: 3(3+1)/2 = 6
For n = 4: 4(4+1)/2 = 10
Now, let's calculate the differences between consecutive terms:
For n = 1 to n = 2: 3 - 1 = 2
For n = 2 to n = 3: 6 - 3 = 3
For n = 3 to n = 4: 10 - 6 = 4
Looking at the differences, we can see that the differences between consecutive terms are not constant. Therefore, none of the options provided (a. 4th differences, b. 3rd differences, c. 2nd differences, d. 1st differences) represent a constant value of level differences for this polynomial.
So, the answer is none of the above.
To determine which level differences of the polynomial n(n+1)/2 are a constant value, we need to find the differences between consecutive terms and observe their patterns.
First, let's compute the polynomial values for the input values of n from 1 to 5:
- For n = 1: 1(1+1)/2 = 1
- For n = 2: 2(2+1)/2 = 3
- For n = 3: 3(3+1)/2 = 6
- For n = 4: 4(4+1)/2 = 10
- For n = 5: 5(5+1)/2 = 15
Next, let's find the differences between consecutive terms:
- Difference between the 1st and 2nd term: 3 - 1 = 2
- Difference between the 2nd and 3rd term: 6 - 3 = 3
- Difference between the 3rd and 4th term: 10 - 6 = 4
- Difference between the 4th and 5th term: 15 - 10 = 5
Observing the pattern of the differences, we can see that the values are increasing by one with each step. This means that the first differences (the differences between consecutive terms) form an arithmetic sequence with a common difference of 1.
Based on this observation, we can conclude that the 1st differences of the polynomial n(n+1)/2 are a constant value. Therefore, the answer is option d. 1st differences.