If the first term is 3, the second is 6, the third is 7, the fourth is -2, what is the nth term?

Question

trying to determine formula and determine the nth term

Answer 1

Only 4 points have been given: 3, 6, 7, -2, so there may be many ways to derive the equation which calculates the n-th term.

Here's one method. First note that this is not a linear progression. Let's assume a cubic progression:

T(n)=an³+bn²+cn+d where a, b, c, d are constant coefficients. Let the sequence start with n=0, so T(0)=3=d and:

T(1)=6=a+b+c+d, or a+b+c=3 because we know that d=3.

T(2)=7=8a+4b+2c+d, 8a+4b+2c=4.

T(3)=-2=27a+9b+3c+3, 27a+9b+3c=-5.

T(2)-2T(1)=6a+2b=-2⇒3a+b=-1.

T(3)-3T(1)=24a+6b=-14⇒12a+3b=-7.

T(3)-3T(1)-3(T(2)-2T(1))=3a=-4, a=-4/3⇒b=-1-(-4)=3⇒c=3-(-4/3+3)=4/3.

Therefore, T(n)=-4n³/3+3n²+4n/3+3, which can be written T(n)=⅓(-4n³+9n²+4n+9).