 A group of students are learning a long poem to perform at the school concert. Each week they are taught a certain number of verses. The first week they are taught one verse and by the end of the second week they know three verses. At the end of the third week the students can recite six verses and at the end of the fourth week they know ten. How many verses would they be able to recite after 12 weeks?

To determine how many verses the students will be able to recite after 12 weeks, we need to identify the pattern in the number of verses learned each week.

Let's denote the number of verses learned by the students at the end of week \( n \) as \( V(n) \).

We are given:
- \( V(1) = 1 \)
- \( V(2) = 3 \)
- \( V(3) = 6 \)
- \( V(4) = 10 \)

We observe the following differences between consecutive weeks:
- \( V(2) - V(1) = 3 - 1 = 2 \)
- \( V(3) - V(2) = 6 - 3 = 3 \)
- \( V(4) - V(3) = 10 - 6 = 4 \)

These differences (2, 3, 4, ...) suggest an increasing pattern. We can infer that the differences themselves follow an arithmetic sequence with a common difference of 1. To generalize this, the number of verses learned by the end of week \( n \) follows the formula for the \( n \)-th triangular number, which is given by:

\[ V(n) = \frac{n(n+1)}{2} \]

Let's verify this with the given values:
- For \( n = 1 \):
  \[ V(1) = \frac{1(1+1)}{2} = \frac{2}{2} = 1 \]
- For \( n = 2 \):
  \[ V(2) = \frac{2(2+1)}{2} = \frac{6}{2} = 3 \]
- For \( n = 3 \):
  \[ V(3) = \frac{3(3+1)}{2} = \frac{12}{2} = 6 \]
- For \( n = 4 \):
  \[ V(4) = \frac{4(4+1)}{2} = \frac{20}{2} = 10 \]

Since the pattern holds, we can use this formula to find the number of verses learned by the end of week 12:

\[ V(12) = \frac{12(12+1)}{2} = \frac{12 \times 13}{2} = \frac{156}{2} = 78 \]

Therefore, the students will be able to recite 78 verses after 12 weeks.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the general formula for the \( n \)-th triangular number?
2. How can we derive the formula for the sum of the first \( n \) natural numbers?
3. How can we prove that the formula \(\frac{n(n+1)}{2}\) correctly calculates the number of verses?
4. What is the sum of the first 12 triangular numbers?
5. If the sequence continued for 20 weeks, how many verses would the students know?
6. How would the sequence change if the students learned an additional verse each week?
7. Can we find a recurrence relation that defines this sequence?
8. What are the properties of triangular numbers?

Tip: Triangular numbers have a variety of interesting properties and connections to other areas of mathematics, such as combinatorics and geometry.

Explore how to calculate the number of verses students can recite over 12 weeks using the triangular number formula. Understand the pattern in verse learning and its application in arithmetic sequences. Suitable for students in Grades 4-6.

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