Prove the formula for the nth term of an arithmetic sequence.

The formula for the nth term of an arithmetic sequence is an = a1 + (n-1)d.

An arithmetic sequence is a sequence of numbers where each term is obtained by adding a fixed number (d) to the previous term. The first term of an arithmetic sequence is denoted by a1, the second term by a2, and so on. The nth term of an arithmetic sequence is denoted by an.

To derive the formula for the nth term of an arithmetic sequence, we can use the following steps:

1. Write out the first few terms of the sequence to identify the pattern.
For example, consider the arithmetic sequence: 2, 5, 8, 11, 14, ...

2. Write out the general formula for the nth term of the sequence.
Let an = a1 + (n-1)d, where d is the common difference between consecutive terms.

3. Substitute the values of a1 and d into the formula.
In the example sequence, a1 = 2 and d = 3, so an = 2 + (n-1)3.

4. Simplify the formula.
Expand the brackets and simplify to get an = 3n - 1.

Therefore, the formula for the nth term of an arithmetic sequence is an = a1 + (n-1)d, where a1 is the first term and d is the common difference between consecutive terms. This formula can be used to find any term in the sequence, given its position (n) in the sequence.