Question

Write an expression to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term.

6, 12, 18, 24, ...


an =

Answer 1

To describe the given sequence \( 6, 12, 18, 24, \ldots \) using \( n \) to represent the position of a term in the sequence, we can observe that this is an arithmetic sequence where each term increases by 6.

The first term (when \( n = 1 \)) is 6. The common difference (the amount added each time) is also 6.

The general formula for the \( n \)-th term of an arithmetic sequence can be written as:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Where:
- \( a_1 \) is the first term
- \( d \) is the common difference
- \( n \) is the position of the term

For this sequence:
- \( a_1 = 6 \)
- \( d = 6 \)

Plugging these values into the formula gives us:

\[
a_n = 6 + (n - 1) \cdot 6
\]

This simplifies to:

\[
a_n = 6 + 6(n - 1) = 6 + 6n - 6 = 6n
\]

Thus, the expression for the \( n \)-th term of the sequence is:

\[
a_n = 6n
\]