question: find the 7th term of this sequence 2,9,28,65,126,217 ?
The terms in the sequence are:
2 9 28 65 126 127 --- this is the sequence a1, a2, a3, ...
7 19 37 61 91 --- the 1st differences - this is the sequence b1, b2, b3, ...
12 18 24 30 --- the 2nd differences - this is the sequence c1, c2, c3, ...
6 6 6 --- the third differences - this is the sequence d1, d2, d3, ...
Since the third differences are a constant value, then we have a mathematically defined sequence of terms.
We could now establish a formula for the nth term of the sequence, but since all we need is the next term, then we can work that out manually. Working backwards, the 7th term is 344. This is confirmed as shown in the table below
2 9 28 65 126 217 344
7 19 37 61 91 127 --- the 1st differences
12 18 24 30 36 --- the 2nd differences
6 6 6 6 --- the third differences
Answer: the 7th term of the sequence is: 344
p.s. if you need/want it, the expression for the nth term of this sequence is,
an = a1 + (n - 1)b1 +c1/2*(n^2 - 3n + 2) + d/6*(n^3 - 6n^2 + 11n - 6)
where a1 is the 1st term in the sequence.
b1 is the 1st term in the sequence of 1st differences.
c1 is the 1st term in the sequence of 2nd differences.
d is the any term in the sequence of 3rd differences.