Question

What is a8 for the geometric sequence 6,561;−2,187;729;−243;…6,561;−2,187;729;−243;….

Answer 1

Consider a geometric sequence $(u_n)$

let $u_1=a$ and the common ratio be r, then the sequence is constructed as follows:

$u_1, u_2, u_3..... = a, ar, a r^{2}, a r^{3}, ...$

we can observe that each term of the sequence is its previous term * r.


In the given sequence, to find the common ratio we divide 6,561 by −2,187 and get -3. This means that $r= -\frac{1}{3}$

Let the first term $u_1=a=6,561$, 

then the eighth term is  

$u_8=a* r^{7}= 6,561 (- \frac{1}{3} )^{7}= -6,561*( \frac{1}{3} )^{7}= \frac{-6,561}{2187} =-3$


Answer: -3