RIGHT ANSWER=BRAINLIEST AND 5 STARTITLE: Donation for a CauseSITUATION:Suppose you are running a foundation for street children. A rich benefactor offers twooptions in giving personal donation.OPTION A: To give P1000 pesos on day 1, P999 on day 2, P998 on day 3 with the process to end after 1000 daysOPTION B: To give P1 on day 1, P2 on day 2, P3 on day 3 for 1000 days.You have to decide today which option you want to take.Answer the following: 1) Which of the two options results in more money for the foundation? Show your computations.2)Which option are you going to take? Explain your choice.
Answer (1)
To determine which donation option is better, we need to compute the total amount of money each option will generate over 1000 days.OPTION AIn Option A, the donation decreases by 1 peso each day, starting from P1000. The sequence of donations is:- Day 1: P1000- Day 2: P999- Day 3: P998- …- Day 1000: P1This is an arithmetic sequence where:- The first term [tex]a_1[/tex]is P1000- The common difference d is -1- The number of terms [tex]n[/tex]is 1000The sum [tex]S_n[/tex] of an arithmetic sequence is given by:[tex]S_n = \frac{n}{2} \times (2a_1 + (n - 1) \times d)[/tex]Substituting the values:[tex]S_{1000} = \frac{1000}{2} \times (2 \times 1000 + (1000 - 1) \times (-1))[/tex][tex]S_{1000} = 500 \times (2000 - 999)[/tex][tex]S_{1000} = 500 \times 1001[/tex][tex]S_{1000} = 500500[/tex]So, the total donation for Option A is P500,500.OPTION BIn Option B, the donation increases by 1 peso each day, starting from P1. The sequence of donations is:- Day 1: P1- Day 2: P2- Day 3: P3- …- Day 1000: P1000This is also an arithmetic sequence where:- The first term [tex]a_1[/tex] is P1- The common difference d is 1- The number of terms n is 1000The sum [tex]S_n[/tex]of this arithmetic sequence is:[tex]S_n = \frac{n}{2} \times (2a_1 + (n - 1) \times d)[/tex]Substituting the values:[tex]S_{1000} = \frac{1000}{2} \times (2 \times 1 + (1000 - 1) \times 1)[/tex][tex]S_{1000} = 500 \times (2 + 999)[/tex][tex]S_{1000} = 500 \times 1001[/tex][tex]S_{1000} = 500500[/tex]So, the total donation for Option B is also P500,500.DECISION AND EXPLANATION1) Both options result in the same total amount of money: P500,500.2) Since both options result in the same total donation, the choice between Option A and Option B can be based on other factors such as:- Preference for the Donation Pattern: Option A starts with a larger donation and decreases, while Option B starts small and increases. Depending on the immediate need, one pattern might be more suitable.- Sustainability: Option B’s increasing donations might be more aligned with a growing financial strategy, while Option A’s decreasing donations might reflect an immediate boost.Ultimately, the choice could depend on the urgency of the need or preference for how donations are received.
