Given a sequence of numbers: 2, 5, 10, 17, find the formula for the nth term and calculate the 8th term of the sequence.

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Accepted Answer

Step:1 ：The given sequence is 2, 5, 10, 17. We can notice that the difference between the consecutive terms is increasing by 2 each time. So the sequence can be a quadratic sequence, and the general formula for a quadratic sequence is (an^{2} + bn + c).Step:2 ：To find the values of a, b, and c, we can set up a system of equations using the first three terms of the sequence: [begin{cases}a + b + c = 24a + 2b + c = 59a + 3b + c = 10end{cases}]Step:3 ：Solving the system of equations, we get (a = 1), (b = 1), and (c = 0). So the nth term of the sequence is (n^{2} + n).Step:4 ：Substitute n = 8 into the formula (n^{2} + n), we get the 8th term of the sequence.

Answer

Step: 1 ：The given sequence is 2, 5, 10, 17. We can notice that the difference between the consecutive terms is increasing by 2 each time. So the sequence can be a quadratic sequence, and the general formula for a quadratic sequence is (an^{2} + bn + c).Step: 2 ：To find the values of a, b, and c, we can set up a system of equations using the first three terms of the sequence: [begin{cases}a + b + c = 24a + 2b + c = 59a + 3b + c = 10end{cases}]Step: 3 ：Solving the system of equations, we get (a = 1), (b = 1), and (c = 0). So the nth term of the sequence is (n^{2} + n).Step: 4 ：Substitute n = 8 into the formula (n^{2} + n), we get the 8th term of the sequence.

Given a sequence of numbers: 2, 5, 10, 17, find the formula for the nth term and calculate the 8th term of the sequence.

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