The equations given below illustrate a suspected number pattern. Determine the next equation. \( \begin{array}{l}1=1^{2} \\ 1+2+1=2^{2} \\ 1+2+3+2+1=3^{2} \\ 1+2+3+4+3+2+1=4^{2}\end{array} \)

To determine the next equation in the given pattern, let's analyze the existing equations step by step. 1. **First Equation:** \[ 1 = 1^2 \] This is straightforward, as \(1\) is indeed equal to \(1^2\). 2. **Second Equation:** \[ 1 + 2 + 1 = 2^2 \] The left side sums to \(4\), which equals \(2^2\). 3. **Third Equation:** \[ 1 + 2 + 3 + 2 + 1 = 3^2 \] The left side sums to \(9\), which equals \(3^2\). 4. **Fourth Equation:** \[ 1 + 2 + 3 + 4 + 3 + 2 + 1 = 4^2 \] The left side sums to \(16\), which equals \(4^2\). Now, let's identify the pattern in the left-hand side of the equations: - The first equation has \(1\) term. - The second equation has \(3\) terms: \(1, 2, 1\). - The third equation has \(7\) terms: \(1, 2, 3, 2, 1\). - The fourth equation has \(13\) terms: \(1, 2, 3, 4, 3, 2, 1\). The number of terms in each equation appears to follow the pattern of the sequence of odd numbers: - \(1\) (1 term) - \(3\) (2 terms) - \(7\) (3 terms) - \(13\) (4 terms) The next number of terms can be calculated as follows: - The difference between the number of terms is \(2, 4, 6\), which suggests the next difference will be \(8\). - Therefore, the next number of terms will be \(13 + 8 = 21\). Now, let's construct the next equation. The next equation will have \(5\) as the highest number, and it will be structured as follows: \[ 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 \] Now, let's calculate the sum of these terms: \[ 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 \] Finally, we can express this as: \[ 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 5^2 \] Thus, the next equation in the pattern is: \[ 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 5^2 \]