Respuesta :
Answer: total cost to be saved is $2600. Her saving pattern is 2, 4, 8,…
a) The pattern of her saving is in geometric sequence. i.e. a=2, r=4/2=2 0r 8/4=2 ( a = First term, r = common ratio) so, expression for amount saved on day (n) = t(n) = ar^(n-1), where: a = first day of saving r = common ration n = number of day
b) Expression that represents the total amount saved by day (n) (including day n) = S = a(r^n-1)/r-1 where: S = sum of amount saved a = first day of saving r = common ration n = number of day
c) To buy the car, she needs at least $2600 which is equal 260000 cents. S = a(r^n-1)/r-1 = 260000 = 2(2^n-1)/r-1 = 260000/2 = 2^n-1/2-1 = 130000 = 2n^-1 = 130000+1 = 2^n = 130001 = 2^n = n = ln(130001)/ln(2) = n = 16.988
So… for n to satisfy the least value of 130001 cents, n should be at least 17 Therefore it will take at least 17 days for her to save enough money to buy a car
Step-by-step explanation:
It will take her about 17 days to buy the car worth 260000 cents
If a student decides she wants to save money to buy a used car that cost $2600 (260,000 cents)
If she saves 2 cents the first day and doubles the amount thereafter, the sequence of savings will be:
2, 4, 8...
This sequence is geometric in nature
In order to determine how long it will take her to save 260,000, we will use the sum of a GP formula expressed as:
[tex]S_n = \frac{a(r^n-1)}{r-1}[/tex]
Given the folowing
a = 2
r = 4/2 = 8/4 = 2
Sn = 260,000
Substitute into the formula the given parameters
[tex]260000= \frac{2(2^n-1)}{2-1}\\260000/2=2^n-1\\130000 = 2^n - 1\\2^n = 130000 + 1\\2^n = 130001\\nlog 2=log130001\\n = \frac{log130001}{log2} \\n \approx 17[/tex]
This shows that it will take her about 17 days to buy the car worth 260000 cents
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