**Write an explicit formula for ***a*_{n}_{}_{,}** the ***n***th term of the sequence 38, 44, 50,…**

**Final Answer:**

The explicit formula for the *n*th term of the sequence 38, 44, 50,… is *a**n*=32+6*n*. This sequence is an arithmetic sequence with a first term of 38 and a common difference of 6. The general formula can be derived using the standard formula for arithmetic sequences.

**Explanation:**

The sequence given is 38, 44, 50,… which is an arithmetic sequence. An arithmetic sequence is one where each term after the first is found by adding a constant to the previous term. In this case, the first term *a*1=38 and the difference between consecutive terms is *d*=44−38=6 (the common difference).

To find the explicit formula for the *n*th term of the sequence, we use the formula for the *n*th term of an arithmetic sequence:

*a**n*=*a*1+(*n*−1)⋅*d*

Substituting the values we have:

*a**n*=38+(*n*−1)⋅6

To simplify, we expand the expression:

*a**n*=38+6*n*−6

This simplifies to:

*a**n*=32+6*n*

Thus, the explicit formula for the *n*th term of the sequence is *a*_{n}=32+6*n*.

In summary:

- First term:
*a*_{1}=38
- Common difference:
*d*=6
- Explicit formula:
*a*_{n}_{}=32+6*n*

**Examples & Evidence:**

For example, if we want to find the 1st term, we substitute *n*=1: *a*_{1}=32+6(1)=38. If we want to find the 5th term, we substitute *n*=5: *a*_{5}=32+6(5)=32+30=62.

The properties of arithmetic sequences and formulas for finding the nth term are well-established in mathematics. The process of deriving the explicit formula in this case is standard and consistent with established arithmetic sequence definitions.

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