Question

write the recursive formula for the sequence 3, 24, 192, 1536, ..

Answer 1

To find a recursive formula for the sequence 3, 24, 192, 1536, ..., we first observe the pattern in the terms.

Let's denote the sequence by \( a_n \), where:
- \( a_1 = 3 \)
- \( a_2 = 24 \)
- \( a_3 = 192 \)
- \( a_4 = 1536 \)

Now, let's look at the ratio of consecutive terms:

- \( \frac{a_2}{a_1} = \frac{24}{3} = 8 \)
- \( \frac{a_3}{a_2} = \frac{192}{24} = 8 \)
- \( \frac{a_4}{a_3} = \frac{1536}{192} = 8 \)

It appears that each term is obtained by multiplying the previous term by 8. Thus, we can derive a recursive formula as follows:

\[
a_n = 8 \cdot a_{n-1} \quad \text{for } n \geq 2
\]

And we also need to specify the first term:

\[
a_1 = 3
\]

So the complete recursive formula is:

\[
\begin{align*}
a_1 & = 3 \\
a_n & = 8 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{align*}
\]