se inductive reasoning to predict the next line in the sequence of computations. Then se a calculator or perform the arithmetic by hand to determine whether your conjecture s correct. \[ \begin{array}{r} 4+8=\frac{8 \times 3}{2} \\ 4+8+12=\frac{12 \times 4}{2} \\ 4+8+12+16=\frac{16 \times 5}{2} \\ 4+8+12+16+20=\frac{20 \times 6}{2} \end{array} \] The next line in the sequence is \( 4+8+12+16+20+24= \) \( \square \) (Do not simplify. Use integers or fractions for any numbers in the expression.)

To determine the next line in the sequence, let's analyze the pattern using inductive reasoning. **Given Sequence:** \[ \begin{array}{r} 4+8=\frac{8 \times 3}{2} \\ 4+8+12=\frac{12 \times 4}{2} \\ 4+8+12+16=\frac{16 \times 5}{2} \\ 4+8+12+16+20=\frac{20 \times 6}{2} \end{array} \] **Observations:** 1. **Left-Hand Side (LHS):** - Each term added increases by 4. - The number of terms increases by 1 in each subsequent line. - Sequence of sums: - 4 + 8 - 4 + 8 + 12 - 4 + 8 + 12 + 16 - 4 + 8 + 12 + 16 + 20 2. **Right-Hand Side (RHS):** - The pattern on the RHS involves the last term of the LHS multiplied by an incrementing integer, then divided by 2. - For example: - \(8 \times 3\) (since there are 2 terms in LHS) - \(12 \times 4\) (since there are 3 terms in LHS) - \(16 \times 5\) (since there are 4 terms in LHS) - \(20 \times 6\) (since there are 5 terms in LHS) **Predicting the Next Line:** - **LHS:** Adding the next term \(24\) to the existing sum. \[ 4 + 8 + 12 + 16 + 20 + 24 \] - **RHS:** Following the pattern, multiply the last term \(24\) by \(7\) (since there will be 6 terms in the LHS) and divide by \(2\). \[ \frac{24 \times 7}{2} \] **Final Next Line:** \[ 4+8+12+16+20+24= \frac{24 \times 7}{2} \] Answer: \[ 4+8+12+16+20+24= \frac{24 \times 7}{2} \]