- Identify the first term of the sequence as \(a_1 = 36\).
- Calculate the common ratio by dividing the second term by the first: \(r = \frac{-14.4}{36} = -0.4\).
- Verify the common ratio with subsequent terms.
- Write the recursive formula as \(a_1 = \boxed{36}; a_n = a_{n-1} \cdot \boxed{-0.4}\).
Explanation
- Understanding the Problem
The provided geometric sequence is \(36, -14.4, 5.76, -2.304, \ldots\). A geometric sequence is defined by its first term, \(a_1\), and a common ratio, \(r\), where each subsequent term is calculated by multiplying the previous term by the common ratio: \(a_n = a_{n-1} \cdot r\). We are given the sequence and need to identify the first term and the common ratio to construct the recursive formula.
- Finding the First Term and Common Ratio
From the given sequence, the first term is clearly:
\[a_1 = 36\]
To find the common ratio \(r\), we divide the second term by the first term:
\[r = \frac{a_2}{a_1} = \frac{-14.4}{36} = -0.4\]
We can verify this ratio by checking the subsequent terms:
\[\frac{a_3}{a_2} = \frac{5.76}{-14.4} = -0.4\]
\[\frac{a_4}{a_3} = \frac{-2.304}{5.76} = -0.4\]
Since the ratio is consistent, we confirm \(r = -0.4\).
- Formulating the Recursive Rule
With the first term \(a_1 = 36\) and the common ratio \(r = -0.4\), the recursive formula for the geometric sequence is:
\[a_1 = 36; a_n = a_{n-1} \cdot (-0.4)\]
Thus, the first box is 36 and the second box is -0.4. The final result is:
\[a_1 = \boxed{36} ; a_n = a_{n-1} \cdot \boxed{-0.4}\]
Examples
In real life, geometric sequences describe processes with constant percentage change, such as compound interest or the decay of a substance. For example, if a company's investment depreciates by 40% each year, the value of the investment would follow a geometric sequence with a common ratio of \(1 - 0.4 = 0.6\). Alternatively, in a scenario where a radioactive isotope loses 40% of its current mass every hour, the remaining amount of mass at each hour would be modeled by multiplying the previous amount by a factor of 0.6. Understanding these ratios allows for accurate prediction of future states based on current trends.