The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?
The nth term of a geometric sequence is given by $ar^{n-1}$, where a is the first term and r is the common ratio. So the fifth term is $ar^4=7!$ and the eighth term is $ar^7=8!$. Now if we divide these two equations, we get $\frac{ar^4}{ar^7}=\frac{7!}{8!}$. That means $r = 2$. So, if we plug that back into the first equation, we get $a\cdot 2^4 = 7!$. We get $a =\frac{7!}{2^4}$. So the first term is $\frac{7!}{16}$. And since $7!=5040$, the first term is $315$.

