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

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]