Final answer:
Sylvia saves a total of $62 in 5 weeks by doubling her savings each week. She will have enough to purchase a $125 terrarium after 6 weeks following this pattern of saving.
Step-by-step explanation:
Sylvia is saving money using a pattern where she doubles the amount saved each week. To find out how much total money Sylvia will save in 5 weeks, we can create a table where each week's savings are listed:
- Week 1: $2
- Week 2: $4
- Week 3: $8
- Week 4: $16
- Week 5: $32
The total savings after 5 weeks is the sum of these amounts, which is $2 + $4 + $8 + $16 + $32 = $62.
Since Sylvia wants to buy a terrarium that costs $125, we need to continue this pattern until her savings meet or exceed this amount. This sequence represents a geometric series with a first term of $2 and a common ratio of 2. To find out in how many weeks Sylvia will have enough money, we would use the formula for the sum of a geometric series Sn = a(1-rn)/(1-r), where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms. We continue the doubling pattern until the sum is at least $125.
Calculating Weeks for $125
Following the pattern:
After 6 weeks, Sylvia will have saved enough money to buy the terrarium since the total will be $126 which is just over $125.