Question

identify a pattern in the given list of numbers. then use this pattern to find the next number. (more than one pattern might exist, so it is possible that there is more than one correct answer.)
1, 5, 1, 10, 1, 20, 1, ______

the next number is \square

Explanation:

Step1: Analyze odd - numbered terms

Looking at the sequence \(1, 5, 1, 10, 1, 20, 1,\dots\), the terms at positions \(1\) (\(n = 1\)), \(3\) (\(n=3\)), \(5\) (\(n = 5\)), \(7\) (\(n=7\)) are all \(1\). So the pattern for odd - numbered terms (where the position \(n\) is odd, \(n = 1,3,5,7,\dots\)) is that the term is \(1\).

Step2: Analyze even - numbered terms

The terms at positions \(2\) (\(n = 2\)): \(5\), position \(4\) (\(n=4\)): \(10\), position \(6\) (\(n = 6\)): \(20\). Let's see the relationship between these terms. We can observe that \(5\times2=10\), \(10\times2 = 20\). So the pattern for even - numbered terms (where the position \(n\) is even, \(n=2,4,6,\dots\)) is that each term is double the previous even - numbered term.

Step3: Find the next term

The next term is at position \(8\) (which is an even - numbered position). The previous even - numbered term (at position \(6\)) is \(20\). So we multiply \(20\) by \(2\) to get the next term. \(20\times2=40\).

Answer:

\(40\)