Write an explicit formula for a, start subscript, n, end subscripta

Question

n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 37, comma, 30, comma, 23, comma, dot dot dot37,30,23,....

Answer 1

To find the explicit formula for the sequence \( a_n \) given by the terms 37, 30, 23, ..., we first need to identify the pattern in the sequence.

1. Identify the first few terms:

   • \( a_1 = 37 \)
   • \( a_2 = 30 \)
   • \( a_3 = 23 \)

2. Calculate the differences between consecutive terms:

   • \( a_2 - a_1 = 30 - 37 = -7 \)
   • \( a_3 - a_2 = 23 - 30 = -7 \)

This indicates that the sequence is an arithmetic sequence with a common difference of -7.

3. General form of an arithmetic sequence: An arithmetic sequence can be expressed in the form: \[ a_n = a_1 + (n-1) \cdot d \] where \( a_1 \) is the first term and \( d \) is the common difference.

4. Substituting values: Here, \( a_1 = 37 \) and \( d = -7 \): \[ a_n = 37 + (n-1)(-7) \]

5. Simplifying the formula: \[ a_n = 37 - 7(n-1) \] \[ a_n = 37 - 7n + 7 \] \[ a_n = 44 - 7n \]

Thus, the explicit formula for the \( n \)-th term of the sequence is: \[ \boxed{a_n = 44 - 7n} \]