How do we find the formula for quadertic sequences?
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.
It is a simple thong to do. First of all lets have this sequence 3,7,13,21,31. Then we need find something called the second difference since the difference between this sequence is not constant as it is 4.6,8,10 so we find the second difference which is two. Then we half the numer which we found and we put it in front of n^2 so the formula will start with n^2. Then we find using the part of this formula to make a sequence which 1,4,9,16,25 then we find the difference between this sequence and the original one then we record the difference and make a sequence of the difference and then we put the difference between th sequence we did and we put after then squared so it becomes n^2+1 and then we see whats missing to make the sequence correct then you find it