Question

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

Answer 1

Answer:

$x_n=7(-3)^{n-1}$

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:

$x_n=ar^{n-1}$

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:

$a_2=-21\\a_5=567$

Substitute them into the equations:

$x_2=ar^{(2)-1}\\-21=ar$

And:

$a^5=ar^{(5)-1}\\567=ar^4$

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

$r=-\frac{21}{a}$

And substitute this into the second equation:

$567=a(\frac{-21}{a} )^4$

Simplify:

$567=a(\frac{(-21)^4}{a^4})$

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

$\frac{567}{1}=\frac{194481}{a^3}$

Cross multiply:

$194481=567a^3\\a^3=194481/567=343$

Take the cube root of both sides:

$a=\sqrt[3]{343} =7$

Therefore, the initial value is 7.

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

$r=-21/a\\r=-21/(7)\\r=-3$

Thus, the common ratio is -3.

Therefore, the equation is:

$x_n=7(-3)^{n-1}$