anita had $400 in her savings account when she went to college. her parents will add $200 to her account…

anita had $400 in her savings account when she went to college. her parents will add $200 to her account each month.\nmiguel had $25 in his savings account. his parents will double the amount in his account each month.\nif anita and miguel do not take any money from their accounts, whose account will grow faster? explain why.

anita had $400 in her savings account when she went to college. her parents will add $200 to her account each month.\nmiguel had $25 in his savings account. his parents will double the amount in his account each month.\nif anita and miguel do not take any money from their accounts, whose account will grow faster? explain why.

Answer

Explanation:

Step1: Write Anita's account - growth formula

Anita has an initial amount of $a_0 = 400$ and a monthly increase of $d = 200$. Her account - balance formula after $n$ months is a linear function $A(n)=400 + 200n$.

Step2: Write Miguel's account - growth formula

Miguel has an initial amount of $m_0 = 25$ and his amount is doubled each month. His account - balance formula after $n$ months is an exponential function $M(n)=25\times2^n$.

Step3: Compare the growth rates

For linear functions like $A(n)=400 + 200n$, the rate of change is constant (the slope is 200). For exponential functions like $M(n)=25\times2^n$, the rate of change is not constant. As $n$ increases, the difference between consecutive values of $M(n)$ gets larger. For example, when $n = 1$: $A(1)=400+200\times1 = 600$ and $M(1)=25\times2^1 = 50$. When $n = 2$: $A(2)=400 + 200\times2=800$ and $M(2)=25\times2^2=100$. When $n = 3$: $A(3)=400+200\times3 = 1000$ and $M(3)=25\times2^3 = 200$. But as $n$ gets larger, say $n = 6$: $A(6)=400+200\times6=1600$ and $M(6)=25\times2^6=25\times64 = 1600$. And for $n = 7$, $A(7)=400+200\times7 = 2000$ and $M(7)=25\times2^7=25\times128 = 3200$.

Answer:

Miguel's account will grow faster because his account - balance follows an exponential growth model ($M(n)=25\times2^n$), and exponential functions grow faster than linear functions (Anita's account - balance is $A(n)=400 + 200n$) in the long - run.