Question

write an explicit formula for $a_n$, the $n^{\text{th}}$ term of the sequence $14, 10, 6,....$
answer
attempt 1 out of 2

Explanation:

Step 1: Identify the sequence type

The sequence is \(14, 10, 6, \dots\). Check the difference between consecutive terms: \(10 - 14=- 4\), \(6 - 10=-4\). So, it is an arithmetic sequence with first term \(a_1 = 14\) and common difference \(d=-4\).

Step 2: Recall the explicit formula for arithmetic sequence

The explicit formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step 3: Substitute the values of \(a_1\) and \(d\)

Substitute \(a_1 = 14\) and \(d=-4\) into the formula:
\[

$$\begin{align*} a_n&=14+(n - 1)(-4)\\ &=14-4n + 4\\ &=18-4n \end{align*}$$

\]

Answer:

\(a_n=18 - 4n\)