The sequence given is:

$1, 2, 3, 4, 5, \dots$

This is an arithmetic sequence where each term increases by 1. To find a general expression for the $n$-th term $t_n$ of this sequence, we can use the formula for the $n$-th term of an arithmetic sequence:

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

where:

• $a$ is the first term of the sequence,
• $d$ is the common difference between consecutive terms,
• $n$ is the term number.

For this sequence:

• The first term $a = 1$,
• The common difference $d = 1$.

Substituting into the formula:

$t_n = 1 + (n-1) \cdot 1$ $t_n = n$

So, the general expression for the $n$-th term is simply:

$t_n = n$

This means that the $n$-th term of the sequence is just $n$ itself.

Would you like more details or have any questions on this?

Related Questions:

1. What is the formula for the $n$-th term of a geometric sequence?
2. How do you derive the sum of an arithmetic sequence?
3. What is the difference between arithmetic and geometric sequences?
4. How can you find the sum of the first $n$ terms in this sequence?
5. What if the common difference in an arithmetic sequence was a fraction or a negative number?

Tip:

For arithmetic sequences, always check if the difference between consecutive terms is constant. This will help in identifying the sequence type and applying the correct formula!