Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively

Step-by-step explanation:

First, write some equations so we can figure out the common ratio and the initial term. The standard explicit formula for a geometric sequence is:

Where xₙ is the nth term, a is the initial value, and r is the common ratio.

We know that the second and fifth terms are -21 and 567, respectively. Thus:

Substitute them into the equations:

And:

To find a and r, divide both sides by a in the first equation:

And substitute this into the second equation:

Simplify:

The as cancel out. (-21)^4 is 194481:

Cross multiply:

Take the cube root of both sides:

Therefore, the initial value is 7.

And the common ratio is (going back to the equation previously):

Thus, the common ratio is -3.

Therefore, the equation is: