The given sequence is:

$2, 6, 12, 20, 30, 42, \dots$

First, let's analyze the pattern in the sequence. The difference between consecutive terms forms another sequence:

$6 - 2 = 4,\quad 12 - 6 = 6,\quad 20 - 12 = 8,\quad 30 - 20 = 10,\quad 42 - 30 = 12$

This shows that the sequence of differences is $4, 6, 8, 10, 12, \dots$, which is an arithmetic sequence with a first term of 4 and a common difference of 2.

Now, we can express the $n$th term of the original sequence as:

$a_n = 2 + (2 + 4 + 6 + \dots + (2n - 2))$

This sequence $2 + 4 + 6 + \dots + (2n - 2)$ is itself an arithmetic series where:

• First term: $2$
• Common difference: $2$
• Number of terms: $n-1$

The sum of the first $n-1$ terms of this arithmetic sequence can be found using the formula:

$S_{n-1} = \frac{(n-1)}{2} \times \text{(first term + last term)} = \frac{(n-1)}{2} \times \left(2 + 2(n-1)\right)$

Simplifying the expression:

$S_{n-1} = (n-1) \times (n)$

Thus, the $n$th term of the sequence $a_n$ is:

$a_n = 2 + S_{n-1} = 2 + n(n-1)$

Simplifying further:

$a_n = n^2 + n$

Now, to find the sum of the first 50 terms, we use the sum formula for an arithmetic series:

$S_{50} = \sum_{n=1}^{50} a_n = \sum_{n=1}^{50} (n^2 + n)$

This can be split into two sums:

$S_{50} = \sum_{n=1}^{50} n^2 + \sum_{n=1}^{50} n$

Using the sum formulas:

$\sum_{n=1}^{50} n^2 = \frac{50 \times 51 \times 101}{6} = 42925$ $\sum_{n=1}^{50} n = \frac{50 \times 51}{2} = 1275$

Adding them together gives:

$S_{50} = 42925 + 1275 = 44200$

So, the sum of the first 50 terms of the sequence is 44,200.

Would you like any details or have any questions?

Here are 5 related questions to explore:

1. What is the sum of the first 100 terms of this sequence?
2. How would you find the $n$th term if the sequence had a different pattern of differences?
3. Can you generalize the formula for the sum of any arithmetic sequence?
4. What is the sum of squares of the first 50 natural numbers?
5. How does this method change if the sequence was geometric instead of arithmetic?

Tip: When dealing with sequences, look for patterns in differences to determine the type and formula for the sequence.