This problem might be a little hard to understand because it was translated from Spanish, if you understand Spanish the original problem is below.
English:
Succession problem
A father gives his son a choice in which way he wants to receive weekly pay for the following year (52 weeks). The first way is to receive 25 euros a week, and the second way is to start charging 1 euro the first week and each week to collect two euros more than the previous one. How would you collect more money at the end of the year? How much more money?
Spanish:
Problema sobre succesiones
Un padre le da a elegir a su hijo de qué manera quiere recibir la paga semanal durante el siguiente año (52 semanas). La primera forma es recibir 25 euros semanales, y la segunda manera es empezar cobrando 1 euro la primera semana y cada semana ir cobrando dos euros más que la anterior. ¿De qué forma cobraría más dinero al cabo del año?. ¿Cuánto dinero más?
52*25 = ___
52/2 (2*1 + 52*2) = ____
compare
To determine which method will yield more money at the end of the year and how much more money, we can calculate the total amount of money received using each method.
Method 1:
In this method, the son receives 25 euros every week for 52 weeks. To calculate the total amount, simply multiply 25 euros by 52 weeks:
Total amount = 25 euros/week * 52 weeks = 1300 euros
Method 2:
In this method, the son starts by receiving 1 euro in the first week, and then the amount increases by 2 euros every week. We can create a progression to determine the amount received each week:
1, 3, 5, 7, 9, ...
To find the amount received in the last week, we need to find the nth term of this arithmetic progression. The nth term (Tn) of an arithmetic progression can be calculated using the formula:
Tn = a + (n - 1) * d
where a is the first term, n is the number of terms, and d is the common difference.
Here, the first term (a) is 1 euro and the common difference (d) is 2 euros.
Using the formula, we can find the amount received in the last week (n = 52):
Tn = 1 + (52 - 1) * 2 = 1 + 51 * 2 = 1 + 102 = 103 euros
Now, we can calculate the total amount by summing up all the amounts received in each week. We can use the formula for the sum of an arithmetic series to simplify this calculation:
Sum = (n/2)(2a + (n - 1)d)
Here, n = 52 (number of terms), a = 1 (first term), and d = 2 (common difference).
Sum = (52/2)(2*1 + (52 - 1)*2)
Sum = 26(2 + 51*2)
Sum = 26(2 + 102)
Sum = 26 * 104
Sum = 2704 euros
Comparing the two methods:
Method 1: 1300 euros
Method 2: 2704 euros
Therefore, with Method 2, the son would receive more money at the end of the year. The difference in the amount received between the two methods is:
2704 euros - 1300 euros = 1404 euros
So, the son would receive 1404 euros more with Method 2 than with Method 1.