A sequence has constant 3rd differences of 24. What is the coefficient of the highest power term in its polynomial expression?
The constant third differences mean that the degree of the polynomial must be 3. If $a_0,a_1,a_2,$ are the terms of the sequence, then the third differences are \[\begin{array}{c@{~}c@{~}c@{~}c@{~}c@{~}c} &&&&&&a_0&&&&&&\\ &&&&&&\;\;\;\uparrow&&&&&&\\ &&a_1&&-8&&0&&-8&&a_0&&\\ &&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\\ a_2&&-8&&-8&&-8&&-8&&a_0&&-8&&-8&&-8&&a_0 \end{array}\]Since the third differences are constant and there are 4 numbers in the diagonal, we have \[8+4d+6d+4d=24,\]which gives $d=2.$ Thus, the terms of the sequence are \[\begin{array}{c@{~}c@{~}c@{~}c@{~}c@{~}c} &&&&&&a_0&&&&&&\\ &&&&&&\;\;\;\uparrow&&&&&&\\ &&a_1&&2&&0&&2&&a_0&&\\ &&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\\ a_2&&2&&2&&2&&2&&a_0&&2&&2&&2&&a_0 \end{array}\]and checking that this works, we find that $a_n = 2\binom{n}{2}a_0$ works. Thus, the sequence can be expressed as a polynomial: \[2\binom{n}{2}a_0 = a_0\cdot 2\cdot \frac{n(n-1)}{2} = a_0n(n-1).\]Thus, the coefficient of the highest power term is $\boxed{-1}.$
To find the coefficient of the highest power term in the polynomial expression of a sequence with constant 3rd differences, we need to determine the degree of the polynomial.
The constant 3rd differences indicate that the sequence is generated by a cubic polynomial. Therefore, the highest power term will have a power of 3.
The coefficient of the highest power term can be found by multiplying the constant 3rd difference by the factorial of the degree.
In this case, since the constant 3rd difference is 24 and the degree is 3, we have:
Coefficient of highest power term = 24 * (3!) = 24 * 6 = 144.
Therefore, the coefficient of the highest power term in the polynomial expression is 144.
To find the coefficient of the highest power term in the polynomial expression, we need to determine the degree of the polynomial. The degree of a polynomial is equal to one less than the number of terms with non-zero coefficients.
In this case, we have constant 3rd differences, which means that the original sequence is a polynomial of degree 3. The 3rd differences in the sequence indicate that there is a cubic polynomial involved.
To find the coefficient of the highest power term, we can use the formula:
Coefficient of highest power term = (3rd difference) / (3!)
Since the 3rd difference is given as 24, we can substitute it into the formula:
Coefficient of highest power term = 24 / (3!)
The factorial of 3 is equal to 3 × 2 × 1 = 6, so we can simplify the expression:
Coefficient of highest power term = 24 / 6 = 4
Therefore, the coefficient of the highest power term in the polynomial expression is 4.