Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively. Please explain this to me very carefully.

Question

campbell_st · Answer

a term in a geometric series is 

$$T_{n} = ar^{n -1}$$

so you have for the 2nd term

$$-6 =ar$$

and the 5th term 

$$162 = ar^4$$

this can be written as

$$162 = ar \times r^3$$

substitute the information from the 2nd term 

$$162 = -6 \times r^3$$

you can now solve for the common ration r

when you have it, you will be able to solve for the 1st term a. 

hope this helps

anonymous · Answer

Why did we put 162 = ar * r^(3) ?

anonymous · Answer

because we know that $-6 = ar$, so if we substitute it in we will be left with one variable $r$ (the common ratio), and we can solve for $r$

Then after you find $r$ you can substitute it into $-6 = ar$ to find $a$

campbell_st · Answer

because you are then able to find the common ratio... 

you get 

$$-27 = r^3$$

solve for r

and you'll then find the 1st term is a = 2

anonymous · Answer

what  is the explicit rule?