Question

exponential relationships
module goal: students will learn about geometric sequences, exponential functions and exponential models such as exponential growth and decay.
engage penny doubling challenge
read the section below and complete the following activity.
quick, you only have 15 seconds to make one choice from the two options below. this may or may not be a trick question, so choose wisely!
option 1
collect $1,000,000 right now.
option 2
start with a penny ($0.01) and double it every day. collect the money on the 30th day.
directions: circle one of the options above
after circling your initial guess, complete the table below by doubling the value each day (the first three days have been done for you). use the table to calculate the total amount collected in option b above.
start with $0.01 (1 cent) and double it each day to determine the value on the 30th day
day 1 $0.01 x 2 = $0.02 (2 cents)
day 2 $0.02 x 2 = $0.04 (4 cents)
day 3 $0.04 x 2 = $0.08 (8 cents)
day 4
day 5
day 6
day 7
day 8
day 9
day 10
day 11
day 12
day 13
day 14
day 15
day 16
day 17
day 18
day 19
day 20
day 21
day 22
day 23
day 24
day 25
day 26
day 27
day 28
day 29
day 30

Explanation:

Step1: Identify the formula for exponential growth

The amount of money $A$ after $n$ days with an initial amount $a = 0.01$ and a growth - factor of $r = 2$ is given by the formula $A=a\times r^{n - 1}$. Here, $a = 0.01$, $r = 2$, and $n$ is the number of days.

Step2: Calculate the amount on the 30th day

Substitute $a = 0.01$, $r = 2$, and $n = 30$ into the formula $A=a\times r^{n - 1}$. So $A=0.01\times2^{29}$.
We know that $2^{10}=1024$, $2^{29}=2^{3\times10 - 1}=\frac{2^{30}}{2}=\frac{(2^{10})^3}{2}$.
$(2^{10})^3=1024^3=(1000 + 24)^3=1000^3+3\times1000^2\times24 + 3\times1000\times24^2+24^3$.
$1000^3 = 1000000000$, $3\times1000^2\times24=3\times1000000\times24 = 72000000$, $3\times1000\times24^2=3000\times576 = 1728000$, $24^3=24\times24\times24 = 13824$.
$(1000 + 24)^3=1000000000+72000000+1728000+13824=1073741824$.
$\frac{(2^{10})^3}{2}=\frac{1073741824}{2}=536870912$.
$A = 0.01\times536870912=5368709.12$

Answer:

Option 2 is better since the amount of money after 30 days of doubling a penny each day is $\$5368709.12$, which is more than $\$1000000$ in Option 1.