Question

Carla wants to buy a new bike before summer starts in one week. She saves $1 on the first day of the week, $2 on the second day, $4 on thethird day, $8 on the fourth day, and continues this pattern the rest of the week.If the bike costs $150, Carla willenough money to buy the bike before the start of summer.Carla will have

Answer 1

Let's begin by listing out the information given to us:

Time to summer (t) = 1 week = 7 days

First day = $1

Second day = $2

Third day = $4

Fourth day = $8

Fifth day = ?

Sixth day = ?

Seventh day = ?

With the passing of each day, Carla saves double the amount of the previous day. Represented by the geometric progression:

`\begin{gathered} f(x)=2^(x-1) \\ f(1)=2^(1-1)=2^0=1 \\ f(2)=2^(2-1)=2^1=2 \\ f(3)=2^(3-1)=2^2=4 \\ f(4)=2^(4-1)=2^3=8 \\ f(5)=2^(5-1)=2^4=16 \\ f(6)=2^(6-1)=2^5=32 \\ f(7)=2^(7-1)=2^6=64 \end{gathered}`

On the fifth, sixth & seventh day, Carla saved $16, $32 & $64 respectively.

In 1 week (7 days), Carla saved $127

If the bike costs $150, Carla only has $127 of the said sum:

`127-150=-23`

Therefore, Carla needs $23 more to be able to purchase the bike before summer

Hence, Carla will not have enough money to buy the bike before summer