. Choos e the n ext term in the s eque nce 2,5,11,20,32

To find the next term in the sequence, we need to identify the pattern or rule that the sequence follows. Let's examine the differences between consecutive terms:<br /><br />1. Difference between the second term and the first term: \(5 - 2 = 3\)<br />2. Difference between the third term and the second term: \(11 - 5 = 6\)<br />3. Difference between the fourth term and the third term: \(20 - 11 = 9\)<br />4. Difference between the fifth term and the fourth term: \(32 - 20 = 12\)<br /><br />We can observe that the differences between consecutive terms are increasing by 3 each time. This suggests that the sequence follows a quadratic pattern. To find the next term, we need to determine the formula for the nth term of the sequence.<br /><br />Let's assume the sequence follows a quadratic formula of the form \(a_n = an^2 + bn + c\), where \(a\), \(b\), and \(c\) are constants.<br /><br />Using the first three terms of the sequence, we can set up a system of equations:<br /><br />For \(n = 1\):<br />\(a(1)^2 + b(1) + c = 2\)<br />\(a + b + c = 2\)<br /><br />For \(n = 2\):<br />\(a(2)^2 + b(2) + c = 5\)<br />\(4a + 2b + c = 5\)<br /><br />For \(n = 3\):<br />\(a(3)^2 + b(3) + c = 11\)<br />\(9a + 3b + c = 11\)<br /><br />Solving this system of equations, we find that \(a = 1\), \(b = -1\), and \(c = 2\). Therefore, the formula for the nth term of the sequence is \(a_n = n^2 - n + 2\).<br /><br />To find the next term in the sequence, we substitute \(n = 6\) into the formula:<br /><br />\(a_6 = 6^2 - 6 + 2 = 36 - 6 + 2 = 32 + 8 = 40\)<br /><br />Therefore, the next term in the sequence is 40.