20,17,13,8,2,-5 whats the term to term rule

Answer 1

This can only be solved using an interpolating polynomial.

Terms are:

20 , 17 ,13 , 8 , 2 , - 5

17 - 20 = - 3

13 - 17 = - 4

8 - 13 = - 5

2 - 8 = - 6

- 5 - 2 = - 7

The first order differences are:

- 3 , - 4 , - 5 , - 6 , - 7
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- 4 - ( - 3 ) = - 4 + 3 = - 1

- 5 - ( - 4 ) = - 5 + 4 = - 1

- 6 - ( - 5 ) = - 6 + 5 = - 1

- 7 - ( - 6 ) = - 7 + 6 = - 1

Differences of the second order are constant, so the interpolating polynomial is the second degree.

A polynomial of the second degree requires 2 + 1 = 3 points.

Let the points be:

n = 1 , t1 = 20

n = 2 , t2 = 17

n = 3 , t3 = 13

Interpolating polynomial:

tn = a n² + b • n + c

n = 1 , tn = t1

t1= a • 1² + b • 1 + c = 20

a + b + c = 20

n = 2 , tn = t2

t2= a • 2² + b • 2 + c = 17

4 a + 2 b + c = 17

n = 3 , tn = t3

t3= a • 3² + b • 3 + c = 13

9 a + 3 b + c = 13

Now solve system of 3 equations:

a + b + c = 20

4 a + 2 b + c = 17

9 a + 3 b + c = 13

The solution is:

a = - 1 / 2 , b = - 3 / 2 , c = 22

Interpolating polynomial:

tn = a n² + b • n + c

tn = - 1 / 2 n² - 3 / 2 • n + 22

tn = - 1 / 2 ( n² + 3 n - 44 )