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 $n$.