Question

What is a formula for the nth term of the given sequence?
13, comma, 18, comma, 23, dot dot dot
13,18,23...
Answer
Multiple Choice Answers
a, start subscript, n, end subscript, equals, 13, left parenthesis, 5, right parenthesis, start superscript, n, minus, 1, end superscripta
n
​
=13(5)
n−1

a, start subscript, n, end subscript, equals, 13, plus, 5, na
n
​
=13+5n
a, start subscript, n, end subscript, equals, 18, minus, 5, na
n
​
=18−5n
a, start subscript, n, end subscript, equals, 13, plus, 5, left parenthesis, n, minus, 1, right parenthesisa
n
​
=13+5(n−1)

Answer 1

To find the formula for the nth term of the given sequence \(13, 18, 23, \ldots\), we first need to examine the pattern.

The first term is \(13\), the second term is \(18\), and the third term is \(23\).

Notice that:

- From \(13\) to \(18\), we add \(5\).
- From \(18\) to \(23\), we add \(5\) again.

This indicates that the sequence is arithmetic with a common difference of \(5\).

The general formula for the nth term \(a_n\) of an arithmetic sequence can be expressed as:

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

where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

In this case:

- The first term \(a_1\) is \(13\).
- The common difference \(d\) is \(5\).

Thus, the formula for the nth term is:

\[
a_n = 13 + (n - 1) \cdot 5
\]

Now, let's look at the provided options:

1. \( a_n = 13(5)^{n-1} \) – This suggests exponential growth, which does not match our linear sequence.
2. \( a_n = 13 + 5n \) – This gives \(18\) for \(n=1\), which is incorrect.
3. \( a_n = 18 - 5n \) – This decreases as \(n\) increases, which is incorrect.
4. \( a_n = 13 + 5(n - 1) \) – This matches our derived formula.

Thus, the correct answer is:

\[
\boxed{a_n = 13 + 5(n - 1)}
\]