a savings account has a balance of $1. saving...

# Explanation: ## Step1: Define functions for each plan Let $n$ be the number of months. For Plan A, the balance $A(n)=1 + 1000n$. For Plan B, the balance $B(n)=2^n$. ## Step2: Compare for short - run We check the values of $A(n)$ and $B(n)$ for small $n$. For $n = 1$: $A(1)=1+1000\times1 = 1001$ and $B(1)=2^1 = 2$ For $n = 2$: $A(2)=1 + 1000\times2=2001$ and $B(2)=2^2 = 4$ For $n = 3$: $A(3)=1+1000\times3 = 3001$ and $B(3)=2^3 = 8$ For $n = 4$: $A(4)=1+1000\times4=4001$ and $B(4)=2^4 = 16$ For $n = 5$: $A(5)=1+1000\times5 = 5001$ and $B(5)=2^5 = 32$ For $n = 6$: $A(6)=1+1000\times6=6001$ and $B(6)=2^6 = 64$ For $n = 7$: $A(7)=1+1000\times7 = 7001$ and $B(7)=2^7 = 128$ For $n = 8$: $A(8)=1+1000\times8=8001$ and $B(8)=2^8 = 256$ For $n = 9$: $A(9)=1+1000\times9 = 9001$ and $B(9)=2^9 = 512$ For $n = 10$: $A(10)=1+1000\times10=10001$ and $B(10)=2^{10}=1024$ For $n = 11$: $A(11)=1+1000\times11 = 11001$ and $B(11)=2^{11}=2048$ For $n = 12$: $A(12)=1+1000\times12=12001$ and $B(12)=2^{12}=4096$ For $n = 13$: $A(13)=1+1000\times13 = 13001$ and $B(13)=2^{13}=8192$ For $n = 14$: $A(14)=1+1000\times14=14001$ and $B(14)=2^{14}=16384$ So Plan A is better for the first 13 months. ## Step3: Analyze for long - run The function for Plan A is a linear function $y = 1000n+1$ and the function for Plan B is an exponential function $y = 2^n$. Exponential functions grow faster than linear functions as $n$ gets large. So Plan B is better in the long - run. # Answer: a. Savings Plan A is better in the short run because it results in more money than the other plan for the first 13 months. b. Savings Plan B is better in the long run because exponential functions ($B(n)=2^n$) grow faster than linear functions ($A(n)=1 + 1000n$) as the number of months $n$ increases.