Answer:
[tex]x_n=7(-3)^{n-1}[/tex]
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:
[tex]x_n=ar^{n-1}[/tex]
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:
[tex]a_2=-21\\a_5=567[/tex]
Substitute them into the equations:
[tex]x_2=ar^{(2)-1}\\-21=ar[/tex]
And:
[tex]a^5=ar^{(5)-1}\\567=ar^4[/tex]
To find a and r, divide both sides by a in the first equation:
[tex]r=-\frac{21}{a}[/tex]
And substitute this into the second equation:
[tex]567=a(\frac{-21}{a} )^4[/tex]
Simplify:
[tex]567=a(\frac{(-21)^4}{a^4})[/tex]
The as cancel out. (-21)^4 is 194481:
[tex]\frac{567}{1}=\frac{194481}{a^3}[/tex]
Cross multiply:
[tex]194481=567a^3\\a^3=194481/567=343[/tex]
Take the cube root of both sides:
[tex]a=\sqrt[3]{343} =7[/tex]
Therefore, the initial value is 7.
And the common ratio is (going back to the equation previously):
[tex]r=-21/a\\r=-21/(7)\\r=-3[/tex]
Thus, the common ratio is -3.
Therefore, the equation is:
[tex]x_n=7(-3)^{n-1}[/tex]
