The series $2 + 5 + 7 + \dots + 77$ is an arithmetic sequence, where:

• The first term $a = 2$
• The common difference $d = 3$ (since $5 - 2 = 3$)
• The last term $l = 77$

To find the sum of an arithmetic sequence, we use the formula:

$S_n = \frac{n}{2} \cdot (a + l)$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a$ is the first term.
• $l$ is the last term.

First, we need to find $n$, the number of terms in the sequence. This is calculated using the formula for the $n$-th term of an arithmetic sequence:

$l = a + (n - 1) \cdot d$

Substituting the known values:

$77 = 2 + (n - 1) \cdot 3$

Solving for $n$:

$77 - 2 = (n - 1) \cdot 3$ $75 = (n - 1) \cdot 3$ $n - 1 = 25$ $n = 26$

Now that we know $n = 26$, we can substitute it into the sum formula:

$S_{26} = \frac{26}{2} \cdot (2 + 77)$ $S_{26} = 13 \cdot 79$ $S_{26} = 1027$

Thus, the sum of the series is $1027$.

Do you want details or have any questions?

Here are 5 related questions for further exploration:

1. What if the first term was 3 instead of 2?
2. How do you calculate the sum of a geometric series?
3. What is the formula for the sum of an arithmetic series with an unknown last term?
4. How does the common difference affect the growth of an arithmetic sequence?
5. Can you derive the sum formula for an arithmetic sequence from first principles?

Tip: Always confirm whether a sequence is arithmetic by checking that the difference between consecutive terms remains constant.