Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term.

1, 5, 25, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

1 answer

To describe the sequence 1, 5, 25, ..., we first observe the relationship between the terms. The first term \( a_1 = 1 \), the second term \( a_2 = 5 \), and the third term \( a_3 = 25 \).

We can see that:

  • \( a_2 = 5 = 1 \times 5^1 \)
  • \( a_3 = 25 = 5 \times 5^1 = 5^2 \)

It appears that each term can be expressed as a power of 5.

We can see that the terms can be described as:

  • For \( n = 1 \): \( a_1 = 5^{0} = 1 \)
  • For \( n = 2 \): \( a_2 = 5^{1} = 5 \)
  • For \( n = 3 \): \( a_3 = 5^{2} = 25 \)

This suggests that the general formula for the nth term in the sequence can be expressed as: \[ a_n = 5^{n-1} \]

Therefore, the equation that describes the sequence is: \[ a_n = 5^{n-1} \]