The given sequence is -1, 2, 7, 14, 23, ... We are asked to find a formula for the $n$-th term of this sequence.
2025/6/25
1. Problem Description
The given sequence is -1, 2, 7, 14, 23, ... We are asked to find a formula for the -th term of this sequence.
2. Solution Steps
Let the sequence be denoted by .
The first differences are:
The second differences are:
Since the second differences are constant, the sequence can be represented by a quadratic expression of the form .
We have:
Subtracting the first equation from the second and the second from the third, we get:
Subtracting the first of these two equations from the second, we get:
, so .
Substituting into , we get:
, so .
Substituting and into , we get:
, so .
Thus, the formula for the -th term is .