Answer:
a(n) = 2n² -n +2
Explanation:
You want the explicit expression for the n-th term of the sequence that begins 3, 8, 17, 30, 47.
Differences
The first differences of the numbers in the sequence are ...
5, 9, 13, 17
The second differences are ...
4, 4, 4
The constant second differences tell us the sequence is of 2nd degree.
Equation
If d0 is the first term of the sequence, d1 is the first of the first differences, and d2 is the first of the second differences, the equation can be written as ...
a(n) = d0 +(n -1)(d1 +(n -2)/2(d2))
a(n) = 3 +(n -1)(5 +(n -2)/2(4)) = 3 +(n -1)(5 +2n -4) = 3 +(n -1)(2n +1)
a(n) = 3 +(2n² -n -1)
a(n) = 2n² -n +2 . . . . . the n-th term of the sequence
__
Additional comment
The differences tell us the sequence is quadratic, so we can use the quadratic regression function of a calculator or spreadsheet to find the equation. That is shown in the attachment. We have also shown that this equation reproduces the given sequence, and that the next term is 68.
In the above equation for a(n), each factor of the "continued product" is of the form (n -k)/k(dk + .... This can be extended to whatever degree is necessary to model a given sequence. Term numbers are assumed to start with 1 in this model.
<95151404393>