The given sequence is -1, 2, 7, 14, 23, ... We are asked to find a formula for the $n$-th term of this sequence.

AlgebraSequencesSeriesQuadratic SequencesFinding the nth termAlgebraic Manipulation
2025/6/25

1. Problem Description

The given sequence is -1, 2, 7, 14, 23, ... We are asked to find a formula for the nn-th term of this sequence.

2. Solution Steps

Let the sequence be denoted by ana_n.
The first differences are:
2(1)=32 - (-1) = 3
72=57 - 2 = 5
147=714 - 7 = 7
2314=923 - 14 = 9
The second differences are:
53=25 - 3 = 2
75=27 - 5 = 2
97=29 - 7 = 2
Since the second differences are constant, the sequence can be represented by a quadratic expression of the form an=An2+Bn+Ca_n = An^2 + Bn + C.
We have:
a1=A(1)2+B(1)+C=A+B+C=1a_1 = A(1)^2 + B(1) + C = A + B + C = -1
a2=A(2)2+B(2)+C=4A+2B+C=2a_2 = A(2)^2 + B(2) + C = 4A + 2B + C = 2
a3=A(3)2+B(3)+C=9A+3B+C=7a_3 = A(3)^2 + B(3) + C = 9A + 3B + C = 7
Subtracting the first equation from the second and the second from the third, we get:
3A+B=33A + B = 3
5A+B=55A + B = 5
Subtracting the first of these two equations from the second, we get:
2A=22A = 2, so A=1A = 1.
Substituting A=1A = 1 into 3A+B=33A + B = 3, we get:
3(1)+B=33(1) + B = 3, so B=0B = 0.
Substituting A=1A = 1 and B=0B = 0 into A+B+C=1A + B + C = -1, we get:
1+0+C=11 + 0 + C = -1, so C=2C = -2.
Thus, the formula for the nn-th term is an=n22a_n = n^2 - 2.

3. Final Answer

an=n22a_n = n^2 - 2