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# Finding a solution to the following conditions?

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Problem Detail:

So for these 5 conditions, I am trying to find the solution/formula for them. What would $a_n$ equal basically? If it helps, the recurrence relation these 5 conditions were generated from was $a_n = a_{n - 1} + 2n$. Any help would be greatly appreciated.

\begin{align*} a_0 &= 4 \\ a_1 &= 6 \\ a_2 &= 10 \\ a_3 &= 16 \\ a_4 &= 24 \end{align*}

###### Answered By : Yuval Filmus

There are infinitely many sequences starting $4,6,10,16,24$. If I understand you correctly, this sequence was generating according to the rule $a_n = a_{n-1} + 2n$ with the initial value $a_0 = 4$. In that case, you have $$a_n = a_0 + \sum_{k=1}^n (2k) = a_0 + 2 \frac{n(n+1)}{2} = n^2 + n + 4.$$
The text passage presents a mathematical problem involving a sequence defined by a recurrence relation and initial conditions, followed by a solution provided by another user.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
World's most popular travel blog for travel bloggers.

# Finding a solution to the following conditions?

, ,
Problem Detail:

So for these 5 conditions, I am trying to find the solution/formula for them. What would $a_n$ equal basically? If it helps, the recurrence relation these 5 conditions were generated from was $a_n = a_{n - 1} + 2n$. Any help would be greatly appreciated.

\begin{align*} a_0 &= 4 \\ a_1 &= 6 \\ a_2 &= 10 \\ a_3 &= 16 \\ a_4 &= 24 \end{align*}

###### Answered By : Yuval Filmus

There are infinitely many sequences starting $4,6,10,16,24$. If I understand you correctly, this sequence was generating according to the rule $a_n = a_{n-1} + 2n$ with the initial value $a_0 = 4$. In that case, you have $$a_n = a_0 + \sum_{k=1}^n (2k) = a_0 + 2 \frac{n(n+1)}{2} = n^2 + n + 4.$$
user
The text passage presents a mathematical problem involving a sequence defined by a recurrence relation and initial conditions, followed by a solution provided by another user.

TYPE: SINGLE_QUESTION_WITH_ANSWER