Question

Write an explicit formula for a, start subscript, n, end subscripta
n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 25, comma, 20, comma, 15, comma, dot dot dot25,20,15,....

Answer 1

To derive an explicit formula for the \( n \)-th term \( a_n \) of the sequence 25, 20, 15, ..., we first identify the pattern in the sequence.

Observing the terms:
- The first term \( a_1 = 25 \)
- The second term \( a_2 = 20 \)
- The third term \( a_3 = 15 \)

We can see that each term in the sequence decreases by 5. Hence, we have a linear sequence where each term is generated by subtracting a constant difference of 5 from the previous term.

The general formula for the \( n \)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1) \cdot d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

In this case:
- \( a_1 = 25 \)
- \( d = -5 \) (because the sequence is decreasing)

Substituting these values into the formula gives us:
\[
a_n = 25 + (n - 1)(-5)
\]
Simplifying this expression:
\[
a_n = 25 - 5(n - 1)
\]
\[
a_n = 25 - 5n + 5
\]
\[
a_n = 30 - 5n
\]

Thus, the explicit formula for the \( n \)-th term of the sequence is:
\[
\boxed{30 - 5n}
\]