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**Find the sum of the first \( n \)-terms:**
\[ 1 + 5 + 13 + 29 + 61 + \cdots \]

### Analysis of the Sequence:
The sequence provided is: \( 1, 5, 13, 29, 61, \dots \).

We notice that this sequence grows quite fast, and the differences between consecutive terms are:
- \( 5 - 1 = 4 \)
- \( 13 - 5 = 8 \)
- \( 29 - 13 = 16 \)
- \( 61 - 29 = 32 \)

The differences are powers of 2: \( 4, 8, 16, 32, \dots \). This suggests that the sequence follows a quadratic pattern.

### Formula for the Sequence:
We assume the general term of the sequence is quadratic, i.e., of the form:
\[
T_n = an^2 + bn + c
\]
We can determine the values of \( a \), \( b \), and \( c \) using the first few terms of the sequence:

1. For \( T_1 = 1 \), we have:
\[
a(1)^2 + b(1) + c = 1 \quad \Rightarrow \quad a + b + c = 1
\]

2. For \( T_2 = 5 \), we have:
\[
a(2)^2 + b(2) + c = 5 \quad \Rightarrow \quad 4a + 2b + c = 5
\]

3. For \( T_3 = 13 \), we have:
\[
a(3)^2 + b(3) + c = 13 \quad \Rightarrow \quad 9a + 3b + c = 13
\]

Now, I will solve this system of equations to find \( a \), \( b \), and \( c \).The solution to the system of equations is:
\[
a = 2, \quad b = -2, \quad c = 1
\]
Thus, the general term of the sequence is:
\[
T_n = 2n^2 - 2n + 1
\]

### Sum of the First \( n \)-terms:
The sum of the first \( n \)-terms of a sequence is given by:
\[
S_n = \sum_{n=1}^{n} T_n
\]
Where \( T_n = 2n^2 - 2n + 1 \). I'll calculate the general formula for the sum \( S_n \).It seems I attempted to use a symbolic variable in a way that requires an actual integer. Instead, I'll apply the summation formula to find the sum of the sequence's first \( n \) terms using calculus or algebraic methods. Let me correct that.The sum of the first \( n \)-terms of the sequence is given by the formula:
\[
S_n = \frac{n(2n^2 + 1)}{3}
\]

This formula allows you to calculate the sum of the first \( n \)-terms of the sequence.

### Do you have any further questions or would you like additional details?

Here are 5 related questions you can explore:
1. How can this sum formula be derived directly using summation rules?
2. Can you calculate the sum for a specific \( n \), say \( n = 5 \)?
3. How would the formula change if the sequence started with a different first term?
4. What if the differences between consecutive terms followed another pattern, such as cubes?
5. How can we visualize the growth of this sequence?

**Tip:** Whenever you encounter a sequence, looking at the differences between consecutive terms is a powerful way to guess the pattern!

Find the sum of the first n terms: 1 + 5 + 13 + 29 + 61 + ... n terms.

Learn how to find the sum of the first n terms of a quadratic sequence: 1 + 5 + 13 + 29 + 61 + ... The solution includes the formula for the nth term and a method for summing the sequence. Suitable for high school students (Grades 10-12).

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