Let x subscript 1 equals 2 comma space x subscript n equals 3 plus x subscript n minus 1 comma space end subscript n greater or equal than 2. Find a formula for the sequence c deﬁned by c subscript n equals stack sum x subscript i with i equals 1 below and n on top.

We are given the recurrence relation for the sequence \( x_n \):

\[
x_1 = 2, \quad x_n = 3 + x_{n-1}, \quad n \geq 2
\]

The goal is to find a formula for the sequence \( c_n \), where

\[
c_n = \sum_{i=1}^{n} x_i
\]

### Step 1: Analyze the Sequence \( x_n \)

To better understand the sequence \( x_n \), let’s compute the first few terms:

- For \( n = 1 \), \( x_1 = 2 \).
- For \( n = 2 \), \( x_2 = 3 + x_1 = 3 + 2 = 5 \).
- For \( n = 3 \), \( x_3 = 3 + x_2 = 3 + 5 = 8 \).
- For \( n = 4 \), \( x_4 = 3 + x_3 = 3 + 8 = 11 \).

It seems that \( x_n \) follows a linear pattern where each term increases by 3. In fact, the general form of \( x_n \) can be derived from the recurrence relation:

\[
x_n = 2 + (n-1) \cdot 3 = 3n - 1
\]

### Step 2: Write the Formula for \( c_n \)

Now, we need to find the formula for \( c_n \), which is the sum of the first \( n \) terms of the sequence \( x_i \):

\[
c_n = \sum_{i=1}^{n} x_i = \sum_{i=1}^{n} (3i - 1)
\]

We can split the sum into two parts:

\[
c_n = \sum_{i=1}^{n} (3i) - \sum_{i=1}^{n} 1
\]

The first sum is the sum of an arithmetic sequence with the first term \( 3 \), the common difference \( 3 \), and \( n \) terms. The second sum is simply \( n \), as we are summing \( 1 \) \( n \) times.

Let’s evaluate each sum:

\[
\sum_{i=1}^{n} 3i = 3 \sum_{i=1}^{n} i = 3 \cdot \frac{n(n+1)}{2} = \frac{3n(n+1)}{2}
\]

\[
\sum_{i=1}^{n} 1 = n
\]

Thus, we have:

\[
c_n = \frac{3n(n+1)}{2} - n = \frac{3n(n+1) - 2n}{2} = \frac{n(3n + 1)}{2}
\]

### Final Answer:
The formula for \( c_n \) is:

\[
c_n = \frac{n(3n + 1)}{2}
\]

### Next Steps:
Would you like a breakdown of the steps or further clarification on any point? Here are 5 related questions for exploration:

1. How can we prove the formula for \( x_n \) by induction?
2. How would the sum change if the recurrence relation were different (e.g., \( x_n = 2 + x_{n-1} \))?
3. Can we express the sum \( c_n \) as a polynomial and analyze its behavior for large \( n \)?
4. How would you approach calculating \( c_n \) for a given \( n \) without using the formula?
5. What is the relationship between the sum of the sequence \( c_n \) and the sum of the first \( n \) terms of other sequences, such as geometric sequences?

**Tip:** When dealing with summations, try to split them into simpler sums whenever possible to make the calculations more manageable.

Let x subscript 1 equals 2, x subscript n equals 3 plus x subscript n minus 1, end subscript n greater or equal than 2. Find a formula for the sequence c defined by c subscript n equals sum x subscript i with i equals 1 below and n on top.

This problem explores how to find a formula for a sequence defined by a recurrence relation x_n = 3n - 1. The sequence c_n is defined as the sum of the first n terms of x_n. The solution involves understanding recurrence relations, arithmetic sequences, and summation formulas. Suitable for students in Grades 9-12.

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