What is the pattern rule of

Question

1,4,10,28,82?

Answer 1

• From the given data, a finite number of terms and imply that there is a single solution, when in fact that is never true.
• In this case, the first five terms are given. For the general term, there must be one quartic polynomial that will fit, and there are infinitely many polynomials of higher degree.

 

• Let, u[n] be the n th term of the sequence. This is how that single quartic polynomial is found.

 

• Let, u[n] = an^4 + bn^3+ cn^2 + dn + e.

 

• Now , substitute n = 1 through 5 to get a system of five linear equations.

 

• a + b + c + d + e = 1

16a + 8b + 4c + 2d + e = 4

81a + 27b + 9c + 3d + e = 10

256a + 64b + 16c + 4d + e = 28

625a + 125b + 25c + 5d + e = 82.

 

• Solve the system for the five unknowns.

 

• a = 5/8

b = -19/4

c = 115/8

d = -65/4

e = 7.

 

• u[n] = (1/8)(5n^4 - 38n^3 + 115n^2 - 130n + 56).

 

• To see that the five terms fit infinitely many polynomials of higher degree, note that any real number may be supposed for the sixth term. The same method could then be applied to find the coefficients of a quintic polynomial.