ARITHMETIC SEQUENCE
2. The number of residents being vaccinated per day in NCR follows arithmetic progression. On the average, 24 people are being vaccinated a day. If the number of vaccinated residents on the 15th day is 2,020, how many of them will be vaccinated on the 30th day?

3. The government project "Libreng Sakay" during the pandemic benefited Filipino commuters all over the country. If the number of benefited commuters from the program reaches 300,000 on the 5th month of the program and 720,000 on its 14th month, how many commuters will be benefited on the 19th month? (Assuming that there is a common difference in the increase of commuters per month)

4. The MRT train has 12 carriages. The first carriage can carry 100 passengers, the second carriage carries 115 passengers, the third 130 passengers, and so on in an arithmetic sequence. How many passengers can the last carriage carry?

5. One private school in Makati has the following enrollees: 120 students in grade one, 200 students in grade two, 280 students in grade 3 and so on. If the school accepted students until grade 12 and the number of enrollees follows an arithmetic sequence, how many students are in grade 12?


Let's address each arithmetic sequence problem step by step.

### 2. Number of Residents Vaccinated (15th day = 2020)
In this arithmetic sequence, the average number of vaccinated residents per day is 24, and on the 15th day, 2,020 people were vaccinated.

We need to find the number of people vaccinated on the 30th day.

The formula for the *n*-th term in an arithmetic sequence is:

\[
a_n = a_1 + (n-1) \cdot d
\]

Where:
- \( a_n \) is the number of people vaccinated on the *n*-th day.
- \( a_1 \) is the number of people vaccinated on the first day (unknown).
- \( d \) is the common difference (unknown).
- \( n \) is the day number.

We are given:
- \( a_{15} = 2020 \), so \( 2020 = a_1 + 14 \cdot d \)
- The average is 24, which implies:

\[
\text{Average} = \frac{a_1 + a_{30}}{2} = 24
\]

First, let's solve for \( a_1 \) and \( d \).

### 3. Number of Commuters Benefited (5th and 14th months)
In this sequence, we are given:
- The number of commuters on the 5th month, \( a_5 = 300,000 \).
- The number of commuters on the 14th month, \( a_{14} = 720,000 \).
- We need to find the number of commuters on the 19th month, \( a_{19} \).

The formula for the *n*-th term is again:

\[
a_n = a_1 + (n-1) \cdot d
\]

We have:
- \( a_5 = a_1 + 4 \cdot d = 300,000 \)
- \( a_{14} = a_1 + 13 \cdot d = 720,000 \)

We will solve these two equations to find \( a_1 \) and \( d \), then calculate \( a_{19} \).

### 4. Passengers in the Last Carriage (MRT Train)
The number of passengers in each carriage forms an arithmetic sequence, with:
- First carriage: 100 passengers.
- Second carriage: 115 passengers.
- Third carriage: 130 passengers.
- We need to find the number of passengers in the 12th carriage.

The formula is the same as:

\[
a_n = a_1 + (n-1) \cdot d
\]

Where:
- \( a_1 = 100 \)
- \( d = 115 - 100 = 15 \)

We need to find \( a_{12} \).

### 5. Students in Grade 12
The enrollment in each grade forms an arithmetic sequence:
- Grade 1: 120 students.
- Grade 2: 200 students.
- Grade 3: 280 students.
- We need to find the number of students in Grade 12.

Here:
- \( a_1 = 120 \)
- \( d = 200 - 120 = 80 \)

We need to find \( a_{12} \).

---

Let's now solve these equations.Here are the solutions for each arithmetic sequence problem:

### 2. Number of Residents Vaccinated on the 30th Day
- The first day vaccination count \(a_1 = 57,908\) and the common difference \(d = -3,992\).
- The number of people vaccinated on the 30th day \(a_{30} = -57,860\), which is not practical in this context. This indicates there might be an issue with the given conditions, especially the average vaccination rate.

### 3. Number of Commuters Benefited on the 19th Month
- The first month \(a_1 = 113,333.33\) and the common difference \(d = 46,666.67\).
- The number of commuters benefited on the 19th month is approximately \(953,333.33\) commuters.

### 4. Passengers in the 12th Carriage
- The number of passengers in the 12th carriage is \(265\).

### 5. Students in Grade 12
- The number of students in Grade 12 is \(1,000\).

Would you like any further details or clarifications on these calculations?

### Follow-up Questions:
1. How would the results change if the average vaccination rate were different?
2. Can you find the sum of passengers across all carriages in the MRT train?
3. What if the number of commuters increased by a variable rate instead of a fixed one?
4. How many students would there be from grades 1 to 12 combined?
5. Can we generalize this approach to find the total number of vaccinations over a given period?

### Tip:
Always check if the result makes practical sense, especially in real-world scenarios, and adjust the model if needed.

ARITHMETIC SEQUENCE
2. The number of residents being vaccinated per day in NCR follows arithmetic progression. On the average, 24 people are being vaccinated a day. If the number of vaccinated residents on the 15th day is 2,020, how many of them will be vaccinated on the 30th day?

3. The government project 'Libreng Sakay' during the pandemic benefited Filipino commuters all over the country. If the number of benefited commuters from the program reaches 300,000 on the 5th month of the program and 720,000 on its 14th month, how many commuters will be benefited on the 19th month? (Assuming that there is a common difference in the increase of commuters per month)

4. The MRT train has 12 carriages. The first carriage can carry 100 passengers, the second carriage carries 115 passengers, the third 130 passengers, and so on in an arithmetic sequence. How many passengers can the last carriage carry?

5. One private school in Makati has the following enrollees: 120 students in grade one, 200 students in grade two, 280 students in grade 3 and so on. If the school accepted students until grade 12 and the number of enrollees follows an arithmetic sequence, how many students are in grade 12?

Learn how to solve arithmetic sequence problems involving vaccinations, commuter benefits, train capacity, and school enrollment. Discover the step-by-step approach to calculate terms in an arithmetic progression using key formulas. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation