Given the sequence 5, 8, 14, 23, 35, ..., find the nth term of this sequence.
Answer
Solving this system, we find that a = 1, b = 2, and c = 2. Therefore, the nth term of the sequence is \(n^2 + 2n + 2\).
Step 1 ：First, notice that the difference between consecutive terms is increasing by 1 each time: 8 - 5 = 3, 14 - 8 = 6, 23 - 14 = 9, 35 - 23 = 12, and so on. This suggests that the nth term of the sequence is of the form \(an^2 + bn + c\) for some constants a, b, and c.
Step 2 ：To find these constants, we can set up a system of equations using the first three terms of the sequence. For n = 1, 2, 3, we get: a + b + c = 5, 4a + 2b + c = 8, and 9a + 3b + c = 14.
Step 3 ：Solving this system, we find that a = 1, b = 2, and c = 2. Therefore, the nth term of the sequence is \(n^2 + 2n + 2\).
