Geometric And Arithmetic Sequences
a store manager plans to offer discounts on some sweater acording to this sequence: $48, $36, $27, $20.255,... What would be the price of a sweater after 8 discounts? Is this an arithmetic or geometric sequence?
is there a common difference? -12,-9,... Not an AP
is there a common ratio? 3/4, 3/4, 3/4, ... Looks like a GP
So, now the question is, is $48 the original price, or the price after one discount?
One you know that, you can use (3/4)^7 or (3/4)^8
To determine the price of a sweater after 8 discounts, we need to figure out the pattern in the given sequence. By observing the numbers $48, $36, $27, $20.255, we can see that each subsequent number is a fraction of the previous one.
To find out if this sequence is an arithmetic or geometric sequence, we need to check if the ratio between consecutive terms is constant. Let's calculate the ratio between each pair of consecutive terms:
1st term / 2nd term = $48 / $36 = 1.33
2nd term / 3rd term = $36 / $27 = 1.33
3rd term / 4th term = $27 / $20.255 ≈ 1.33
Since the ratio between consecutive terms is constant (approximately 1.33), we can conclude that this sequence is a geometric sequence.
To find the price of a sweater after 8 discounts, we can use the formula for calculating the nth term of a geometric sequence:
nth term = a * r^(n-1)
Where:
a = first term
r = common ratio
n = number of terms
In this case:
a = $48 (the first term)
r = 1.33 (the common ratio)
n = 8 (number of terms)
Now we can substitute these values into the formula:
8th term = $48 * 1.33^(8-1)
After evaluating this expression, we find that the price of a sweater after 8 discounts would be approximately $109.55.
In summary, the price of a sweater after 8 discounts would be approximately $109.55, and the given sequence is a geometric sequence.