Use inductive reasoning to predict the next line in the sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct. \[ \begin{aligned} 3+6 & =\frac{6 \times 3}{2} \\ 3+6+9 & =\frac{9 \times 4}{2} \\ 3+6+9+12 & =\frac{12 \times 5}{2} \\ 3+6+9+12+15 & =\frac{15 \times 6}{2} \end{aligned} \] The next line in the sequence is \( 3+6+9+12+15+18= \) \( \square \) \( \square \). (Do not simplify. Use integers or fractions for any numbers in the expression.)

To predict the next line in the sequence, we first observe the pattern in the previous computations. 1. The left side of each equation is a sum of consecutive multiples of 3: - The first line sums \(3\) and \(6\). - The second line sums \(3\), \(6\), and \(9\). - The third line sums \(3\), \(6\), \(9\), and \(12\). - The fourth line sums \(3\), \(6\), \(9\), \(12\), and \(15\). 2. The right side of each equation follows the pattern: - The first line gives \(\frac{6 \times 3}{2}\). - The second line gives \(\frac{9 \times 4}{2}\). - The third line gives \(\frac{12 \times 5}{2}\). - The fourth line gives \(\frac{15 \times 6}{2}\). From this, we can see that: - The last number in the sum on the left side increases by \(3\) each time (from \(6\) to \(9\) to \(12\) to \(15\)). - The multiplier on the right side also increases by \(1\) each time (from \(3\) to \(4\) to \(5\) to \(6\)). Following this pattern, the next line should be: \[ 3 + 6 + 9 + 12 + 15 + 18 = \frac{18 \times 7}{2} \] Now, let's compute the right side: \[ \frac{18 \times 7}{2} \] Calculating \(18 \times 7\): \[ 18 \times 7 = 126 \] Now, divide by \(2\): \[ \frac{126}{2} = 63 \] Thus, the next line in the sequence is: \[ 3 + 6 + 9 + 12 + 15 + 18 = \frac{18 \times 7}{2} = 63 \] So, the completed next line is: \[ 3 + 6 + 9 + 12 + 15 + 18 = \frac{18 \times 7}{2} \]