Question

Write a recursive formula for a, start subscript, n, end subscripta
n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 1, comma, 5, comma, 25, comma, 125, comma, dot dot dot1,5,25,125,....

Answer 1

The sequence given is \(1, 5, 25, 125, \ldots\). We can see that each term in the sequence can be expressed as a power of 5:

- The first term \(a_1 = 5^0 = 1\)
- The second term \(a_2 = 5^1 = 5\)
- The third term \(a_3 = 5^2 = 25\)
- The fourth term \(a_4 = 5^3 = 125\)

From this pattern, we can derive the explicit formula for the \(n\)-th term:

\[
a_n = 5^{n-1}
\]

To express this as a recursive formula, we can identify the relationship between consecutive terms:

\[
a_n = 5 \cdot a_{n-1} \quad \text{for } n \geq 2
\]

This shows that each term is 5 times the previous term. Additionally, we need to specify the base case:

\[
a_1 = 1
\]

Putting this all together, the recursive formula for the sequence is:

\[
\begin{align*}
a_1 & = 1 \\
a_n & = 5 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{align*}
\]