Question

allen sells 20 trash bags after week one. each subsequent week, allen sells 4% more than the previous week.
christian sells 40 trash bags after week one. each subsequent week, christian sells 35 trash bags.
given the scenario above, if each salesperson continues the pace forever, who will sell more trash bags?
show your work here
○ christian
○ allen
○ neither

Explanation:

Step1: Calculate Allen's total sales

Allen's sales form an infinite geometric series with first term $a_1=20$ and common ratio $r=1+0.04=1.04$. The sum of an infinite geometric series is $\frac{a_1}{1-r}$ when $|r|<1$.
$$ S_{\text{Allen}} = \frac{20}{1-1.04} $$
Wait, correction: $r=1.04$, so $1-r=1-1.04=-0.04$, but since we are summing positive terms, we use the formula for a growing infinite series? No, wait, no—actually, for an infinite geometric series where $r>1$, the sum diverges to infinity. Wait, no, wait: 4% more each week means each term is larger than the previous, so the total sum grows without bound, approaching infinity.

Step2: Calculate Christian's total sales

Christian's sales are an infinite arithmetic series: first term $a_1=40$, then each subsequent term is 35. The sum of an infinite arithmetic series with constant non-zero terms (here, after the first term, all terms are 35, so the series is $40 + 35 + 35 + 35 + ...$) also diverges to infinity. Wait, no—wait, no, let's re-examine:

Wait, correction: Allen's sales: week 1:20, week2:20*1.04, week3:20*(1.04)^2, etc. This is a geometric series with $r=1.04>1$, so the partial sums go to infinity as the number of weeks increases.

Christian's sales: week1:40, week2:35, week3:35, week4:35,... So this is $40 + 35(n-1)$ for n weeks. As n approaches infinity, this sum also goes to infinity. But wait, no—wait, no, the rate of growth: Allen's sales grow exponentially, while Christian's grow linearly (constant addition each week after the first). Exponential growth will eventually outpace linear growth, but since we are going "forever," both sums are infinite? No, wait, no—wait, no, the question says "if each salesperson continues the pace forever, who will sell more trash bags?"

Wait, no, let's recast:

Wait, Allen's total after n weeks: $S_n = 20 \cdot \frac{1.04^n - 1}{1.04 - 1} = 20 \cdot \frac{1.04^n - 1}{0.04} = 500(1.04^n - 1)$

Christian's total after n weeks: $T_n = 40 + 35(n-1) = 5 + 35n$

As $n \to \infty$, $1.04^n$ grows exponentially, so $S_n$ will eventually be larger than $T_n$ and keep increasing faster. But wait, both go to infinity, but the question is asking who sells more "forever"—but actually, the infinite sum for Allen is divergent (goes to infinity), and Christian's is also divergent. But wait, no—wait, no, the problem says "continues the pace forever"—so over an infinite time horizon, both have infinite total sales? But that can't be. Wait, no, maybe I misread:

Wait, the problem says "Allen sells 20 trash bags after week one. Each subsequent week, Allen sells 4% more than the previous week." So week 1:20, week2:20*1.04, week3:20*(1.04)^2, etc. So this is a geometric series with ratio >1, so the sum diverges to infinity.

Christian: "sells 40 trash bags after week one. Each subsequent week, Christian sells 35 trash bags." So week1:40, week2:35, week3:35, week4:35,... So the sum is $40 + 35 + 35 + ... = 40 + 35(\infty) = \infty$.

But that can't be the case. Wait, maybe the question means "who has a higher total in the limit"—but both are infinite. But the options are Christian, Allen, Neither.

Wait, no, wait—wait, maybe I made a mistake with Allen's series. Wait, no: for an infinite geometric series, if $|r|>1$, the sum does not converge, it goes to infinity. For Christian, the series is an infinite arithmetic series with common difference 0 after the first term? No, common difference is -5 from week1 to week2, then 0 after that. So the sum is $40 + 35 + 35 + ... = 40 + 35 \times \infty = \infty$.

But that can't be. Wait, maybe the question is asking about the limit of the total per week? No, the question says "who will sell more trash bags" total, forever.

Wait, no—wait, maybe I misinterpret "each subsequent week, Allen sells 4% more than the previous week"—maybe it's 4% more than the first week? No, the wording is "more than the previous week."

Wait, wait another angle: let's compare the partial sums as n becomes very large.

For Allen: $S_n = 500(1.04^n - 1)$

For Christian: $T_n = 35n + 5$

Take n=100: $1.04^{100} \approx 50.5049$, so $S_{100}=500(50.5049-1)=500*49.5049=24752.45$

$T_{100}=35*100+5=3505$

n=200: $1.04^{200} \approx (1.04^{100})^2 \approx 50.5049^2 \approx 2550.74$, so $S_{200}=500(2550.74-1)=500*2549.74=1,274,870$

$T_{200}=35*200+5=7005$

So Allen's total will be vastly larger than Christian's as n increases, and since we are going forever, Allen's total will be infinitely larger (in the sense that the limit of the ratio $S_n/T_n$ is infinity).

Wait, but the options include "Neither"—but that would be if both are equal, but they are not. Wait, no, maybe the question is asking about the total if we consider the infinite sum, but both are infinite. But in standard math, when two infinite sums are both divergent to infinity, but one grows faster, we can say that the exponential one will eventually be larger and stay larger.

Wait, let's recheck the problem statement: "if each salesperson continues the pace forever, who will sell more trash bags?"

So the answer is Allen, because his sales grow exponentially, so over an infinite time horizon, his total sales will be infinitely greater than Christian's (who has linear growth after the first week).

Answer:

Allen