Question

What is the recursive formula for the sequence 8, 10, 12.5, 15.625, …?

Answer 1

To derive the recursive formula for the sequence 8, 10, 12.5, 15.625, …, let's first examine the ratios between consecutive terms:

1. The first term is \( a_1 = 8 \).
2. The second term is \( a_2 = 10 \). The ratio can be calculated as:
\[
\frac{a_2}{a_1} = \frac{10}{8} = 1.25
\]

3. The third term is \( a_3 = 12.5 \). The ratio is:
\[
\frac{a_3}{a_2} = \frac{12.5}{10} = 1.25
\]

4. The fourth term is \( a_4 = 15.625 \). The ratio is:
\[
\frac{a_4}{a_3} = \frac{15.625}{12.5} = 1.25
\]

The ratio between each consecutive term is consistently \( 1.25 \). This indicates that each term can be obtained by multiplying the previous term by \( 1.25 \).

Thus, we can express the recursive formula as follows:

\[
a_n = 1.25 \cdot a_{n-1} \quad \text{for } n \geq 2
\]

And we need to also specify the first term:

\[
a_1 = 8
\]

Therefore, the complete recursive formula for the sequence is:

\[
\begin{align*}
a_1 & = 8 \\
a_n & = 1.25 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{align*}
\]