Question

write the recursive formula for the sequence 10, 60, 360, 2160, ..

Answer 1

To find the recursive formula for the sequence 10, 60, 360, 2160, ..., we first need to examine the relationship between the terms.

Let's denote the sequence as \( a_n \), where:
- \( a_1 = 10 \)
- \( a_2 = 60 \)
- \( a_3 = 360 \)
- \( a_4 = 2160 \)

Next, we will look for a pattern in how each term relates to its predecessor.

Calculating the ratios of successive terms:
- \( \frac{a_2}{a_1} = \frac{60}{10} = 6 \)
- \( \frac{a_3}{a_2} = \frac{360}{60} = 6 \)
- \( \frac{a_4}{a_3} = \frac{2160}{360} = 6 \)

It appears that each term is multiplied by 6 to get the next term. Therefore, we can express this relationship recursively.

The recursive formula for the sequence can be written as:
\[
a_n = 6 \cdot a_{n-1} \quad \text{for } n \geq 2
\]
with the initial condition:
\[
a_1 = 10
\]

Thus, the recursive formula for the given sequence is:
\[
\begin{align*}
a_1 & = 10, \\
a_n & = 6 \cdot a_{n-1} \quad \text{for } n \geq 2.
\end{align*}
\]