Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively.

Answer 1

Answer:

b. an = 2 • (-3)^(n - 1)

Step-by-step explanation:

Before we test a solution or two, we can easily discard most of them.

We see the values alternate of signs (-5 for the 2nd term and +162 for the 5th term)... so the progression factor has to be negative (in order to alternate sign).  That already excludes answers A and C.

Normally, a geometric progression has the (n-1) exponent, not (n+1), so our chances seem to be better with B.

We can test both D and B with n = 2, to obtain -6

Let's test answer D before:

$a_{2} = 2 * (-3)^{2+1} = 2 * (-3)^{3} = 2 * -27 = -54$

The result is -54, not -6... so it's not the right result.

Let's test answer B then:

$a_{2} = 2 * (-3)^{2-1} = 2 * (-3)^{1} = 2 * -3 = -6$

Right!  Let's verify with n=5 to get 162:

$a_{5} = 2 * (-3)^{5-1} = 2 * (-3)^{4} = 2 * 81 = 162$

Confirmed, answer is B. an = 2 • (-3)^(n - 1)