Question

Write an explicit formula for the following sequence. Find the fifteenth term.
4​, 5​, 6​, 7​, 8​, ...
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Part 1
What is an explicit formula for the given​ sequence?
a Subscript nequals
  
enter your response here
​(Simplify your​ answer.)
Answer
simplify the answer
Answered by GPT-4o mini
The given sequence is: 4, 5, 6, 7, 8, ...

This is an arithmetic sequence where the first term \( a_1 = 4 \) and the common difference \( d = 1 \).

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

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

Substituting the values into the formula:

\[
a_n = 4 + (n - 1) \cdot 1
\]

Simplifying this:

\[
a_n = 4 + n - 1 = n + 3
\]

So the explicit formula for the given sequence is:

\[
a_n = n + 3
\]

Now, to find the fifteenth term (\( n = 15 \)):

\[
a_{15} = 15 + 3 = 18
\]

Thus, the fifteenth term is:

\[
\boxed{18}
\]
Answered by GPT-4o mini
The explicit formula derived from the sequence \(4, 5, 6, 7, 8, \ldots\) is:

\[
a_n = n + 3
\]

This formula is already in its simplest form.

To summarize:
- The explicit formula for the sequence is:

\[
\boxed{n + 3}
\]

- The fifteenth term of the sequence is:

\[
a_{15} = 15 + 3 = 18
\]

Thus, the answer for the fifteenth term is:

\[
\boxed{18}
\]

If you were looking for a different kind of simplification or further detail, please let me know!