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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!