You should remember that in quadratic sequences, the common difference between two terms changes, but if you look at the difference between two differences, this is constant.If this property occurs in a sequence of terms, it means that the sequence is a quadratic sequence.
You should look at the example 7,16,31,52,...
Evaluate the difference between two consecutive terms:
16-7 = 9
31-16 = 15
52 - 31 = 21
You need to evaluate the second difference between two consecutive differences such that:
15 - 9 = 6
21 - 15 = 6
Hence, the quadratic sequence is + 3n^2.
You need to subtract 3n^2 from 7,16,31,52,... to determine the residue such that:
Hence, evaluating the formula of quadratic sequence yields T=3n^2 + 4.
A quadratic sequence is a series of terms that respect the following:
1^2, 2^2, 3^2, 4^2, ......, n^2, ....
We'll consider n terms of the quadratic sequence and we'll calculate their sum:
1^2 + 2^2 + 3^2 + ... + n^2 = S (1)
We know that the formula for the squares of the first terms of the sum is:
S = n(n+1)(2n+1)/6
Now, we can have any number of terms of the quadratic series, we'll also be able to determine their sum.