In the two sequences {an} {BN}, an ＞ 0, BN ＞ 0, and an, BN ^ 2, an + 1 are equal difference sequences, and BN ^ 2, an + 1, BN + 1 ^ 2 are equal ratio sequences A1 = 1, B1 = root 2 find Sn = 1 / A1 + 1 / A2 +... + 1 / an

In the two sequences {an} {BN}, an ＞ 0, BN ＞ 0, and an, BN ^ 2, an + 1 are equal difference sequences, and BN ^ 2, an + 1, BN + 1 ^ 2 are equal ratio sequences A1 = 1, B1 = root 2 find Sn = 1 / A1 + 1 / A2 +... + 1 / an

This problem can be proved by mathematical induction as follows:
Suppose BK = (K + 1) / radical 2 a (K + 1) = (K + 1) (K + 2) / 2
It is easy to get the expression BK + 1 = a (K + 1) / BK from the proportional relation of the problem
The conclusion is BN = (n + 1) / radical 2
By induction, we can get the expression of an as an = n (n + 1) / 2
Then Sn can be obtained by the method of split term
1 / an = [1 / N - 1 / (n + 1)] * 2, then n terms are added to get Sn = 2n / (n + 1)