1. The original price of the product, using arithmetic progression formulas, is <u>7.00 Euros</u> per unit.
2. The total amount that the entrepreneur will be paid for the production if the price of the last product (180th) is 4 times higher than the original price, is <u>1,585 Euros</u>.
<h3>What is arithmetic progression?</h3>
Arithmetic progression is a progression in which every term after the first adds a constant value, called the common difference (d).
We can use the arithmetic progression formulas to find the nth term of a progression as well as the sum of the progression.
The formula for arithmetic progression is:
aₙ = a₁ + (n-1) d
aₙ = the nᵗʰ term in the sequence
a₁ = the first term in the sequence
d = the common difference between terms
Likewise, the formula for the sum of an arithmetic progression is Sₙ = n/2[2a + (n − 1) × d], where, a = first term of arithmetic progression, n = number of terms in the arithmetic progression and d = common difference.
<h3>Data and Calculations:</h3>
The fixed price for the first 160 products = 160x
The price for the 161st product = is 8.50 Euros
The price for each subsequent product more than the previous one = 50 cent
Total products manufactured = 180 products
Total earnings of the entrepreneur = 1,385 Euros
Total earnings, 1,385 Euros = 160x + 265 Euros
= 1,385 = 160x + 265
160x = 1,385 - 265
160x = 1,120
x = 7 Euros.
<h3>Manual Illustration:</h3>
Product 161st 162nd 163rd 164th 165th 166th 167th 168th 169th 170th
Prices 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00
Product 171st 172nd 173rd 174th 175th 176th 177th 178th 179th 180th
Prices 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00
The sum of the prices from the 161st product to the 180th product is <u>265 Euros</u>.
<h3>Using Arithmetic Progression Formula:</h3>
The sum of an arithmetic progression is Sn = n/2[2a + (n − 1) × d]
Where:
a = first term of arithmetic progression (8.50 euros)
n = number of terms in the arithmetic progression (20)
d = common difference (0.50)
Therefore, the value is:
20/2{2 x8.50 + (20 - 1) x 0.50}
= 10 {17 + 9.50}
= 265 Euros
<h3>Assumption 2:</h3>
If the price of the last product is 4 times higher than the original price, the price of the 180th product will be <u>28 Euro</u>s (7 x 4).
The additional amount on the price = 10 Euros (28 - 18)
Additional amount for the additional 20 products = 200 (20 x 10)
The total amount for the additional 20 products (161 to 180) = 465 Euros (265 + 200)
Fixed amount based on the original price for 160 products = 1,120.
The total earnings = <u>1,585 Euros</u> (1,385 + 200) or (465 + 1,120)
<h3>Arithmetic Progression:</h3>
Sn = n/2[2a + (n − 1) × d]
20/2{2 x 18.50 + (20 - 1) x 0.50}
= 10 {37 + 9.50}
= 465 Euros
Learn more about arithmetic progression at brainly.com/question/6561461
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