Write an explicit formula for an, the nth term of the sequence 38, 44, 50,…
Final Answer:
The explicit formula for the th term of the sequence 38, 44, 50,… is n=32+6n. 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 1=38 and the difference between consecutive terms is =44−38=6 (the common difference).
To find the explicit formula for the th term of the sequence, we use the formula for the th term of an arithmetic sequence:
n=a1+(n−1)⋅d
Substituting the values we have:
n=38+(n−1)⋅6
To simplify, we expand the expression:
n=38+6n−6
This simplifies to:
n=32+6n
Thus, the explicit formula for the th term of the sequence is n=32+6n.
In summary:
- First term: 1=38
- Common difference: =6
- Explicit formula: n=32+6n
Examples & Evidence:
For example, if we want to find the 1st term, we substitute =1: 1=32+6(1)=38. If we want to find the 5th term, we substitute =5: 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.