an= n² + 2n -1 find the sum of first six term

To find the sum of the first six terms of the sequence given by \( a_n = n^2 + 2n - 1 \), we need to calculate each term from \( a_1 \) to \( a_6 \) and then add them together.

### Step-by-Step Solution

1. **Find each term from \( a_1 \) to \( a_6 \):**

\[
a_1 = 1^2 + 2 \cdot 1 - 1 = 1 + 2 - 1 = 2
\]

\[
a_2 = 2^2 + 2 \cdot 2 - 1 = 4 + 4 - 1 = 7
\]

\[
a_3 = 3^2 + 2 \cdot 3 - 1 = 9 + 6 - 1 = 14
\]

\[
a_4 = 4^2 + 2 \cdot 4 - 1 = 16 + 8 - 1 = 23
\]

\[
a_5 = 5^2 + 2 \cdot 5 - 1 = 25 + 10 - 1 = 34
\]

\[
a_6 = 6^2 + 2 \cdot 6 - 1 = 36 + 12 - 1 = 47
\]

2. **Calculate the sum of the first six terms:**

\[
S_6 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6
\]

\[
S_6 = 2 + 7 + 14 + 23 + 34 + 47
\]

3. **Compute the sum:**

\[
S_6 = 2 + 7 = 9
\]
\[
S_6 = 9 + 14 = 23
\]
\[
S_6 = 23 + 23 = 46
\]
\[
S_6 = 46 + 34 = 80
\]
\[
S_6 = 80 + 47 = 127
\]

### Final Answer
The sum of the first six terms is \( \boxed{127} \).

Would you like more details or have any questions? 

Here are 5 related questions you might find interesting:
1. What is the general formula to find the sum of any number of terms in a quadratic sequence?
2. How do you find the \(n\)-th term in a quadratic sequence?
3. Can we derive a direct formula to calculate the sum of the first \(n\) terms for this sequence?
4. How does the quadratic term affect the behavior of the sequence?
5. What are some examples of other sequences that grow quadratically?

**Tip:** For quadratic sequences, knowing how to derive the sum formula can save time for larger \(n\).

an = n² + 2n - 1 find the sum of first six terms

Find the sum of the first six terms of the quadratic sequence an = n² + 2n - 1 with a detailed step-by-step solution. This problem focuses on summation of quadratic sequences, ideal for students in Grades 9-10.

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