what is the explicit formula for the arithmetic sequence $\frac{4}{5},\frac{29}{30},\frac{17}{15},\frac{13}{1…

Answer:

C. $a_{n}=\frac{4}{5}+\left(-\frac{1}{6}\right)(n - 1)$

Explanation:

Step1: Recall arithmetic - sequence formula

The explicit formula for an arithmetic sequence is $a_{n}=a_{1}+d(n - 1)$, where $a_{1}$ is the first - term and $d$ is the common difference.

Step2: Identify the first - term

The first - term $a_{1}$ of the sequence $\frac{4}{5},\frac{29}{30},\frac{17}{15},\frac{13}{10}$ is $a_{1}=\frac{4}{5}$.

Step3: Calculate the common difference

$d=a_{2}-a_{1}$, where $a_{2}=\frac{29}{30}$ and $a_{1}=\frac{4}{5}=\frac{24}{30}$. Then $d=\frac{29}{30}-\frac{24}{30}=\frac{5}{30}=\frac{1}{6}$. Also, we can check with other consecutive terms. For example, $a_{3}=\frac{17}{15}=\frac{34}{30}$, $a_{2}=\frac{29}{30}$, and $a_{3}-a_{2}=\frac{34}{30}-\frac{29}{30}=\frac{5}{30}=\frac{1}{6}$. But since the sequence is decreasing, the common difference $d =-\frac{1}{6}$.

Step4: Write the explicit formula

Substitute $a_{1}=\frac{4}{5}$ and $d =-\frac{1}{6}$ into the formula $a_{n}=a_{1}+d(n - 1)$, we get $a_{n}=\frac{4}{5}+\left(-\frac{1}{6}\right)(n - 1)$.