A geometric sequence is defined by the explicit formula an = 5(-3)n-1.

What is the recursive formula for the nth term of this sequence?



A.an = 5an-1
B.an+1 = -3an
C.an+1 = 5an
D.an = -3an-1
E.an+1 = -3an



I think it is A?? am I right?

Technically, only A and D give a formula for the nth term of a sequence. The other three give a formula for the (n+1)st term.

A and C are equivalent, and neither one is the correct formula.
B and E are identical. A typo?
That leaves D.

Proof:

From the formula,

a_n = 5 (-3)^(n-1)
a_(n-1) = 5 (-3)^((n-1) - 1) = 5 (-3)^(n-2)

Manipulating the formula for a_n,

a_n = 5(-3)^(n-2+1)
  = 5 (-3)^(n-2) * (-3)^1
  = 5 (-3)^(n-2) * (-3)

But as determined earlier, a_(n-1) = 5 (-3)^(n-2). Therefore,

a_n = (-3) a_(n-1)

which is D.