this was posted recently, and it will haunt me and my boyfriend until I understand it...

I can't have you and your boy friend being haunted for ever! LOL

$$\\a_n=a_1 r^{n-1}\\\\
a_9=29625\\
a_{14}=145\\
$find $t_{18}\\\\$$

You do this with simultaneous equations

$$\\a_1r^{13}=145\qquad so \qquad a_1=\frac{145}{r^{13}}\\\\
a_1r^{8}=29625\qquad so \qquad a_1=\frac{29625}{r^{8}}\\so\\
\frac{29625}{r^{8}}=\frac{145}{r^{13}}\\\\
\frac{r^{13}}{r^{8}}=\frac{145}{29625}\\\\
\frac{r^{13}}{r^{8}}=\frac{145}{29625}\\\\
r^5=\frac{145}{29625}\\\\
r^5=\frac{29}{5925}\\\\
r\approx 0.3451\qquad $correct to 4 decimal places$\\\\\\
a_1=\frac{29625}{r^{8}}\\\\
a_1\approx \frac{29625}{0.3451^{8}}\\\\
a_1\approx 147 264 709\\\\$$

check

$${\mathtt{147\,264\,709}}{\mathtt{\,\times\,}}{{\mathtt{0.345\: \!1}}}^{{\mathtt{8}}} = {\mathtt{29\,624.999\: \!922\: \!546\: \!973\: \!785\: \!9}}$$   that is near enough to 29625

$${\mathtt{147\,264\,709}}{\mathtt{\,\times\,}}{{\mathtt{0.345\: \!1}}}^{{\mathtt{13}}} = {\mathtt{145.005\: \!058\: \!994\: \!920\: \!558\: \!8}}$$    and that is near enough to 145  

Nice job, Melody.....I did this one and got the same answers as you, but the results seemed so screwy that I thought I was making mistakes....LOL!!!

Yes that is why I went throug the checking proceedure.  I don't normally do that I usually figure that the asker can check for themselves.  :))   Thanks for the points :)