The image shows a mathematical sequence problem that reads:

"1. A sequence $a_1, a_2, a_3, \ldots$ has $a_1 = 2$ and satisfies

$a_{n+1} = \frac{a_n(a_n + 1)}{2}$

for all positive integers $n$. For which values of $n$ are all the terms of the sequence still integers?"

Solution

To determine for which values of $n$ all the terms of the sequence are integers, we start by calculating the first few terms:

1. For $n = 1$: $a_2 = \frac{a_1(a_1 + 1)}{2} = \frac{2(2 + 1)}{2} = \frac{2 \times 3}{2} = 3$ So, $a_2 = 3$, which is an integer.

2. For $n = 2$: $a_3 = \frac{a_2(a_2 + 1)}{2} = \frac{3(3 + 1)}{2} = \frac{3 \times 4}{2} = 6$ So, $a_3 = 6$, which is an integer.

3. For $n = 3$: $a_4 = \frac{a_3(a_3 + 1)}{2} = \frac{6(6 + 1)}{2} = \frac{6 \times 7}{2} = 21$ So, $a_4 = 21$, which is an integer.

4. For $n = 4$: $a_5 = \frac{a_4(a_4 + 1)}{2} = \frac{21(21 + 1)}{2} = \frac{21 \times 22}{2} = 231$ So, $a_5 = 231$, which is an integer.

The pattern suggests that each term is an integer. To generalize, we can observe that if $a_n$ is an integer, $a_{n+1}$ is computed by multiplying $a_n$ and $a_n + 1$, which is always even, and then dividing by 2, thus $a_{n+1}$ remains an integer.

Conclusion:

Given the recursive formula, all terms $a_n$ will be integers for every positive integer $n$. This is because the sequence starts with an integer, and each subsequent term is derived from multiplying an integer with another integer and dividing by 2, which consistently yields an integer.

Would you like more details on this solution or have any further questions?

Follow-up Questions:

1. How would the sequence change if $a_1$ were a different integer?
2. Can you derive a general formula for $a_n$ based on the recursive relation?
3. How does the behavior of this sequence compare to a geometric or arithmetic sequence?
4. What are some real-life applications of such recursive sequences?
5. How does the property of being an integer for all $n$ relate to other types of sequences?

Tip:

When working with recursive sequences, calculating the first few terms often helps in identifying patterns and understanding the general behavior of the sequence.