A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved for the next day. Write the first five terms

Since the soup kitchen starts with 16 gallons and there are 128 ounces in 1 gallong, then the first term of the sequence is ${\displaystyle a_1=16\left(128\right)=2048.}$
If a quarter of the soup is served and the rest is saved for the next day, then three-fourths of the soup is left after each day. The next terms in the sequence can then be found by multiplying each term by $\frac{3}{4}$ to get the next term. The next four terms of the sequence are then:
${\displaystyle a_2=\frac{3}{4}{a}_1=\frac{3}{4}\left(2048\right)=1536}$
${\displaystyle a_3=\frac{3}{4}{a}_2=\frac{3}{4}\left(1536\right)=1152}$
${\displaystyle a_4=\frac{3}{4}{a}_3=\frac{3}{4}\left(1152\right)=864}$
${\displaystyle a_5=\frac{3}{4}{a}_4=\frac{3}{4}\left(864\right)=648}$
The first five terms of the sequence are then 2048,1536,1152,864,648.
Since the terms have a common ratio of ${\displaystyle r=\frac{3}{4}}$, then the sequence is geometric. The formula for the nnth term of a geometric sequence is ${\displaystyle a_n=a_1{r}^n-1}$. Since $a_1=2048$ and $r=34$, then the equation that represents the nnth term of the sequence is ${\displaystyle a_n=2048{\left(\frac{3}{4}\right)}^n-1}$
For the soup to be all gone, you would need an=0 for some value of nn. Multiplying a number by ${\displaystyle \frac{3}{4}}$ repeatedly will give a smaller and smaller number each time you multiply. The number will never equal 0 though, it will just get very close to 0. Therefore, the soup will never be gone.