# 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 of the sequence of the number of fluid ounces of soup left each day. Write an equation that represents the nth term of the sequence. When is all the soup gone?

Question
Sequences
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 of the sequence of the number of fluid ounces of soup left each day. Write an equation that represents the nth term of the sequence. When is all the soup gone?

2021-03-09
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 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 a1=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.

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