the first term of a geometric sequence is 8000 and the fifth term is 500. Determine the common ratio and the sum of the first nine terms​

Answer: The common ratio  = 1/2The sum of 9 terms of GP is 15,968.75Step-by-step explanation:Here, in the given GP:Firs Term a = 8000, Fifth term a(5)  =   500Let Common Ratio = rNow, by the general term of GP:[tex]a_n = a \times (r)^{n-1}[/tex]For, n = 5  [tex]a_5 = a \times (r)^{5-1}[/tex]or, [tex]500  = 8000 \times (r)^{4}\\\implies  (r)^{4} = \frac{500}{8000}  = \frac{1}{16}   = \frac{1}{(2)^4} \\\implies r= \frac{1}{2}[/tex]Hence in the given GP, a   = 8000 and r = 1/2Now, in a  GP sum of n terms is [tex]s_n = \frac{a(1-r^n)}{1-r}[/tex]So, for n = 9, [tex]s_9 = \frac{8000(1- (\frac{1}{2}) ^9)}{1-\frac{1}{2} } = \frac{8000(1- 0.001953)}{0.5 }\\= \frac{8000 \times (0.9980)}{0.5}  =  15,968.75[/tex]So,the sum of 9 terms of GP is 15,968.75.