The sequence provided is:
$25, -5, 1, \dots$

To find the explicit formula for the $n$-th term of the sequence, let's analyze the pattern.

Step 1: Identify if it's an arithmetic or geometric sequence

• To check for an arithmetic sequence, find the common difference $d$ between terms:
  • $-5 - 25 = -30$
  • $1 - (-5) = 6$

The differences are not the same, so it's not arithmetic.

Step 2: Test for a geometric sequence

• To check if it's geometric, find the common ratio $r$:
  • $\frac{-5}{25} = -\frac{1}{5}$
  • $\frac{1}{-5} = -\frac{1}{5}$

Thus, the common ratio $r$ is $-\frac{1}{5}$.

Step 3: Find the general formula for a geometric sequence

The general formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.

In this case:

• $a_1 = 25$
• $r = -\frac{1}{5}$

Therefore, the explicit formula is: $a_n = 25 \cdot \left( -\frac{1}{5} \right)^{n-1}$

This is the formula for the $n$-th term of the sequence.

Would you like more details or have any questions?

Related Questions:

1. How would you calculate the 5th term using the explicit formula?
2. What is the difference between arithmetic and geometric sequences?
3. How would the formula change if the common ratio were positive?
4. What is the sum of the first 5 terms of this sequence?
5. How do you derive a formula for an arithmetic sequence?

Tip:

Always check both the common difference and common ratio to confirm whether a sequence is arithmetic or geometric!