if a boy saved 25 pesos on the first day, 27 pesos on the second day, 29 pesos on third day, and so on. how many days will it take him to save 880 pesos?

The boy is saving money in an arithmetic sequence where the amount saved each day increases by a constant difference. Let's analyze the problem to find out how many days it will take him to save 880 pesos.

### Step-by-Step Solution

1. **Identify the Arithmetic Sequence:**
   - The savings form an arithmetic sequence with:
     - First term \( a = 25 \) pesos (amount saved on the first day).
     - Common difference \( d = 27 - 25 = 2 \) pesos.

2. **General Formula for the Sum of an Arithmetic Sequence:**
   - The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:
   \[
   S_n = \frac{n}{2} \left(2a + (n - 1)d\right)
   \]
   Where:
   - \( n \) = number of terms (days in this context),
   - \( a \) = first term,
   - \( d \) = common difference.

3. **Set Up the Equation:**
   - We need to find the number of days \( n \) such that the total savings \( S_n = 880 \) pesos.
   \[
   880 = \frac{n}{2} \left(2 \times 25 + (n - 1) \times 2\right)
   \]

4. **Simplify the Equation:**
   \[
   880 = \frac{n}{2} \left(50 + 2n - 2\right)
   \]
   \[
   880 = \frac{n}{2} (48 + 2n)
   \]
   \[
   880 = \frac{n}{2} \times 2(n + 24)
   \]
   \[
   880 = n(n + 24)
   \]

5. **Form a Quadratic Equation:**
   \[
   n^2 + 24n - 880 = 0
   \]

6. **Solve the Quadratic Equation:**
   - To solve \( n^2 + 24n - 880 = 0 \), we can use the quadratic formula:
   \[
   n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Where \( a = 1 \), \( b = 24 \), and \( c = -880 \).
   
   Substituting these values into the formula gives:
   \[
   n = \frac{-24 \pm \sqrt{24^2 - 4 \times 1 \times (-880)}}{2 \times 1}
   \]
   \[
   n = \frac{-24 \pm \sqrt{576 + 3520}}{2}
   \]
   \[
   n = \frac{-24 \pm \sqrt{4096}}{2}
   \]
   \[
   n = \frac{-24 \pm 64}{2}
   \]

7. **Calculate the Two Possible Values for \( n \):**
   - \( n_1 = \frac{-24 + 64}{2} = \frac{40}{2} = 20 \)
   - \( n_2 = \frac{-24 - 64}{2} = \frac{-88}{2} = -44 \)

   Since \( n \) represents the number of days, it must be positive. Therefore, \( n = 20 \).

### Conclusion

It will take the boy **20 days** to save 880 pesos.

---

Do you want more details or have any questions?

Here are 5 related questions:
1. What if the boy started saving 30 pesos on the first day and increased the savings by 3 pesos each day?
2. How much would the boy save if he continued this pattern for 30 days?
3. How do you solve a quadratic equation if the discriminant is negative?
4. What is the sum of an arithmetic series formula used for?
5. How would the solution change if the savings increased by 5 pesos each day?

**Tip:** The quadratic formula can solve any quadratic equation of the form \(ax^2 + bx + c = 0\). Always check the discriminant \(b^2 - 4ac\) to determine the nature of the roots.

Learn how many days it takes to save 880 pesos using an arithmetic sequence. Step-by-step solution includes identifying the sequence, applying the sum formula, and solving the resulting quadratic equation. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation