What number comes next in the series? \( 1,2,6,15,31 \)

The sequence is given as \[ 1, \quad 2, \quad 6, \quad 15, \quad 31. \] 1. First, compute the differences between consecutive terms: \[ 2 - 1 = 1, \quad 6 - 2 = 4, \quad 15 - 6 = 9, \quad 31 - 15 = 16. \] 2. Notice that the differences \(1, 4, 9,\) and \(16\) are perfect squares: \[ 1 = 1^2, \quad 4 = 2^2, \quad 9 = 3^2, \quad 16 = 4^2. \] 3. The next difference should then be the square of the next integer: \[ 5^2 = 25. \] 4. Add this difference to the last term in the series: \[ 31 + 25 = 56. \] Thus, the next number in the series is \[ 56. \]