Sequence Challenge: Find $a_{2015}$

In summary, we are given a recursive sequence of nonnegative integers $(a_i)_{i\in \Bbb{N}}$ where $a_2 = 5$, $a_{2014} = 2015$, and $a_n=a_{a_{n-1}}$ for all other positive $n$. By analyzing the pattern in the sequence, we can see that the only possible value of $a_{2015}$ is $2015$. This can be confirmed by plugging in different values for $a_2$ and $a_{2014}$.
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Let $(a_i)_{i\in \Bbb{N}}$ be a sequence of nonnegative integers such that $a_2 = 5$, $a_{2014} = 2015$, and $a_n=a_{a_{n-1}}$ for all other positive $n$. Find all possible values of $a_{2015}$.
 
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Hello! This is an interesting problem. Let's first try to understand the sequence given in the problem. We are given that $a_2 = 5$, which means that $a_5 = a_{a_4} = a_{a_{a_3}} = a_{a_{a_{a_2}}} = a_{a_{a_{a_{a_1}}}} = a_{a_{a_{a_{a_0}}}}$. Since $a_n = a_{a_{n-1}}$ for all positive $n$, we can see that this sequence is recursive in nature.

Now, let's try to find a pattern in the sequence. We can see that $a_3 = a_{a_2} = a_5 = 2015$. Similarly, $a_4 = a_{a_3} = a_{2015}$. Continuing this pattern, we can see that $a_{2015} = a_{a_{a_{a_{...a_5}}}}$, where $a_5$ is repeated $2012$ times. This means that $a_{2015}$ is the $2013$th term in the sequence, which is equal to $2015$.

Therefore, the only possible value of $a_{2015}$ is $2015$. This can also be verified by plugging in different values for $a_2$ and $a_{2014}$.

I hope this helps! If you have any further questions or need clarification, please feel free to ask.
 

FAQ: Sequence Challenge: Find $a_{2015}$

How do you solve a sequence challenge to find a2015?

In order to solve a sequence challenge to find a2015, you will need to know the formula for the sequence and the value of the previous term. Once you have this information, you can plug in the values and use basic algebra to find the value of a2015.

What is the formula for finding the nth term in a sequence?

The formula for finding the nth term in a sequence is usually given in the problem or can be derived from the given terms. It is typically represented as an = a1 + (n-1)d, where a1 is the first term, n is the term you are trying to find, and d is the common difference between terms.

Can you use a calculator to solve a sequence challenge?

Yes, you can use a calculator to solve a sequence challenge. However, it is important to make sure you understand the formula and the steps involved in solving the problem before relying on a calculator. Using a calculator can help with complex calculations, but it is still important to understand the concepts behind the solution.

What if the sequence is not arithmetic or geometric?

If the sequence is not arithmetic or geometric, it may be a more complex sequence such as a Fibonacci sequence or a pattern that requires a different approach to solve. In this case, you will need to carefully examine the given terms and try to identify the pattern or formula that can be used to find a2015.

Are there any shortcuts or tricks for solving sequence challenges?

There are some common tricks or patterns that can be used to solve specific types of sequence challenges, such as the sum of consecutive integers or a pattern involving exponents. However, it is important to understand the underlying concepts and formulas in order to effectively solve any sequence challenge. Practice and familiarity with different types of sequences can also help in finding efficient ways to solve them.

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