An arithmetic sequence is represented by the recursive formula mc011-1.jpg. If the first term in the sequence is 4, write the explicit formula.
The explicit formula for an arithmetic sequence is given by the formula $a_n = a_1 + (n-1)d$, where $a_n$ represents the $n$th term in the sequence, $a_1$ is the first term, $n$ is the position of the term, and $d$ is the common difference between consecutive terms.
So, if the first term in the sequence is 4, our explicit formula becomes:
$a_n = 4 + (n-1)d$
However, we need to find the common difference $d$ to fully write the explicit formula. The given recursive formula is $a_{n} = a_{n-1} + 2$, where $a_{n-1}$ is the previous term.
Using this recursive formula, we can find the common difference $d$ as follows:
$a_n = a_{n-1} + 2$
$4 + (n-1)d = (4 + (n-2)d) + 2$
$4 + nd - d = 4 + nd - 2d + 2$
$0 = d + 2$
$d = -2$
Substituting $d = -2$ into the explicit formula:
$a_n = 4 + (n-1)(-2)$
$a_n = 4 - 2n + 2$
$a_n = 6 - 2n$
Therefore, the explicit formula for the arithmetic sequence with a first term of 4 is $a_n = 6 - 2n$.