Let's analyze each sequence and find the sum of the terms.

1. Sequence: 1, 4, 7, 10, ...

This is an arithmetic sequence with a common difference $d = 3$.

General form: $a_n = a_1 + (n-1) \cdot d = 1 + (n-1) \cdot 3 = 3n - 2$

Sum of the first $n$ terms $S_n$ for an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$

Substitute $a_1 = 1$ and $a_n = 3n - 2$:

$S_n = \frac{n}{2} \cdot (1 + (3n - 2)) = \frac{n}{2} \cdot (3n - 1) = \frac{n(3n - 1)}{2}$

2. Sequence: 3, 5, 7, 9, 11, ...

This is also an arithmetic sequence with a common difference $d = 2$.

General form: $a_n = 3 + (n-1) \cdot 2 = 2n + 1$

Sum of the first $n$ terms: $S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{n}{2} \cdot (3 + (2n + 1)) = \frac{n}{2} \cdot (2n + 4) = n(n + 2)$

3. Sequence: 2, 4, 6, 8, 10, ...

This is another arithmetic sequence with a common difference $d = 2$.

General form: $a_n = 2 + (n-1) \cdot 2 = 2n$

Sum of the first $n$ terms: $S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{n}{2} \cdot (2 + 2n) = n(n + 1)$

4. Sequence: -1, 1, -1, 1, -1, 1, ...

This is an alternating sequence with a repeating pattern.

General form: $a_n = (-1)^{n+1}$

Sum of the first $n$ terms depends on whether $n$ is odd or even:

• If $n$ is even, the sum is 0.
• If $n$ is odd, the sum is $-1$.

5. Sequence: 1, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, ...

This is a harmonic sequence.

General form: $a_n = \frac{1}{n}$

Sum of the first $n$ terms is the harmonic sum: $S_n = \sum_{k=1}^{n} \frac{1}{k}$ This sum does not have a simple closed form like the others.

Recap of Sums

1. $S_n = \frac{n(3n - 1)}{2}$
2. $S_n = n(n + 2)$
3. $S_n = n(n + 1)$
4. Sum is 0 if $n$ is even; $-1$ if $n$ is odd.
5. $S_n$ is the sum of the harmonic series.

Would you like details on any of these sequences, or help with another topic? Here are some related questions:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. Can the sum of an infinite arithmetic sequence be finite?
3. What is the significance of the harmonic series in mathematics?
4. How does the alternating sequence affect its sum over time?
5. Can we approximate the sum of the harmonic series for large $n$?

Tip: The sum of an arithmetic sequence can be quickly calculated by multiplying the number of terms by the average of the first and last term.