Twins, Pierre and Francesca were each given 100 dollars on their 15th bday. they immediately put ther money into their individual money boxes. Each week throughout the next year they added a portion of their weekly pocket money. Puerre added 10 each week. Francesca added 50 cents the first week, $1 the next,$1.50 the next, and so on, addin an extra 50 cents each subsequent week.

Find the amount that each added to their money boxes after 8 weeks

How much did Francesca add to her money box on her 16th bday?

Calculate the total amount they each had in their money boxes after one year

50 + 100 + 150 etc is discrete is an arithmetic series with d = 50 cents

To find the amount each twin added to their money boxes after 8 weeks, we can use the following formulas:

Pierre: Amount added = 10 dollars * Number of weeks
Francesca: Amount added = (0.50 dollars + 0.50 dollars * (week number - 1)) * Number of weeks

Plugging in the values:

Pierre: Amount added = 10 dollars * 8 weeks = 80 dollars
Francesca: Amount added = (0.50 dollars + 0.50 dollars * (8 - 1)) * 8 weeks = (0.50 dollars + 3.50 dollars) * 8 weeks = 4 dollars * 8 weeks = 32 dollars

Therefore, Pierre added 80 dollars and Francesca added 32 dollars to their money boxes after 8 weeks.

To find how much Francesca added to her money box on her 16th birthday, we need to calculate the amount she added in the 52nd week. We can use the formula mentioned earlier:

Francesca: Amount added = (0.50 dollars + 0.50 dollars * (week number - 1)) * Number of weeks

Plugging in the values:

Francesca: Amount added = (0.50 dollars + 0.50 dollars * (52 - 1)) * 52 weeks = (0.50 dollars + 25.50 dollars) * 52 weeks = 26 dollars * 52 weeks = 1,352 dollars

Therefore, Francesca added 1,352 dollars to her money box on her 16th birthday.

To calculate the total amount each twin had in their money boxes after one year, we need to add the initial amount they were given on their 15th birthday to the amount they added throughout the year.

Pierre:
Total amount = Initial amount + Amount added
Total amount = 100 dollars + 10 dollars/week * 52 weeks = 100 dollars + 520 dollars = 620 dollars

Francesca:
Total amount = Initial amount + Amount added
Total amount = 100 dollars + (0.50 dollars + 1 dollar + 1.50 dollars + ... + 26 dollars) * 52 weeks

Now, to calculate the sum of 0.50 dollars to 26 dollars, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)*(first term + last term)
where n is the number of terms and the first term is 0.50 dollars and the last term is 26 dollars:

Sum = (26/2)*(0.50 dollars + 26 dollars)
Sum = 13*(0.50 dollars + 26 dollars)
Sum = 13*(26.50 dollars)
Sum = 344.50 dollars

Total amount = 100 dollars + 344.50 dollars * 52 weeks = 100 dollars + 17,944 dollars = 18,044 dollars

Therefore, Pierre had a total of 620 dollars and Francesca had a total of 18,044 dollars in their money boxes after one year.