Questions: Use inductive reasoning to find the next equation 2^2 - 1^2 = 3 3^2 - 1^2 = 8 5^2 - 2^2 = 21

Solution Steps

To find the next equation using inductive reasoning, we need to identify a pattern in the given equations. Observe the differences between the squares and the results. Once the pattern is identified, apply it to find the next equation.

We start with the given equations: \[ \begin{align_} 2^{2} - 1^{2} &= 3 \\ 3^{2} - 1^{2} &= 8 \\ 5^{2} - 2^{2} &= 21 \end{align_} \] We observe the first numbers \(2\), \(3\), and \(5\) and their corresponding results \(3\), \(8\), and \(21\). The differences between the first numbers suggest a pattern.

The differences between the first numbers are: \[ 3 - 2 = 1 \quad \text{and} \quad 5 - 3 = 2 \] Continuing this pattern, we add \(2\) to the last first number \(5\): \[ 5 + 2 = 7 \]

Using the identified pattern, we calculate the next result: \[ 7^{2} - 2^{2} = 49 - 4 = 45 \]

Final Answer