Polynomial Sequences and Formulas

The video tutorial explains how to use polynomial fitting to find the formula for the nth term of a sequence. It begins by introducing the concept of polynomial fitting and the conditions under which a sequence can be represented by a polynomial. The tutorial then demonstrates how to calculate the first, second, and third differences of a sequence to determine the degree of the polynomial. It proceeds to set up and solve a system of equations to find the coefficients of the polynomial using an augmented matrix. Finally, the closed formula for the sequence is derived, providing a comprehensive understanding of the process.

What is the condition for a sequence to have a closed formula as a degree k polynomial?

• The sequence is periodic.

• The sequence is geometric.

• The sequence is arithmetic.

• The sequence is delta k constant.

What is the first difference of the sequence 3, 6, 12, 22?

• 3, 5, 10

• 3, 6, 9

• 3, 6, 10

• 3, 6, 12

How do you determine the degree of the polynomial for a sequence?

• By finding the first constant sequence of differences.

• By calculating the sum of the sequence.

• By checking if the sequence is arithmetic.

• By finding the product of the sequence.

What is the equation for a sub 0 in terms of a, b, c, and d?

• b = 3

• a = 3

• a + b + c + d = 3

• d = 3

What is the equation derived from a sub 1?

• a + b + c = 3

• a + b + c = 6

• a + b + c = 9

• a + b + c = 12

Which method is used to solve the system of equations for the polynomial coefficients?

• Substitution method

• Trial and error

• Graphical method

• Augmented matrix

What is the value of 'a' in the polynomial equation?

• 1/2

• 1/3

• 1

• 1/6

What is the value of 'b' in the polynomial equation?

• 1

• 2

• 3

• 4

What is the value of 'c' in the polynomial equation?

• 11/6

• 9/6

• 5/6

• 7/6

What is the final closed formula for the sequence?

• 1/2 n^3 + n^2 + 11/6 n + 3

• 1/3 n^3 + n^2 + 11/6 n + 3

• n^3 + n^2 + 11/6 n + 3

• 1/6 n^3 + n^2 + 11/6 n + 3