6. \( 1,3,6,10,15,21, ?, ? \)

The given sequence is: \[ 1, 3, 6, 10, 15, 21, \ ?, \ ? \] Let's analyze the pattern: 1. **Differences Between Terms:** - \(3 - 1 = 2\) - \(6 - 3 = 3\) - \(10 - 6 = 4\) - \(15 - 10 = 5\) - \(21 - 15 = 6\) The differences between consecutive terms are increasing by **1** each time: \(2, 3, 4, 5, 6, \dots\) 2. **Next Differences:** - The next difference after \(6\) would be \(7\). - The following difference would be \(8\). 3. **Calculating the Next Terms:** - **First Missing Term:** \(21 + 7 = 28\) - **Second Missing Term:** \(28 + 8 = 36\) 4. **Triangular Numbers Interpretation:** - The sequence represents the **triangular numbers**, where the \(n\)th term is given by: \[ T_n = \frac{n(n+1)}{2} \] - For \(n = 7\): \[ T_7 = \frac{7 \times 8}{2} = 28 \] - For \(n = 8\): \[ T_8 = \frac{8 \times 9}{2} = 36 \] **Therefore, the next two numbers in the sequence are 28 and 36.**