Sequence, mathematical induction, function, inequality F (x) = x - (3 / 2) x & sup2;, A1 ∈ (0,1 / 2), a (n + 1) = f (an), n ∈ n * An

Sequence, mathematical induction, function, inequality F (x) = x - (3 / 2) x & sup2;, A1 ∈ (0,1 / 2), a (n + 1) = f (an), n ∈ n * An

It is proved that the function f (x) = x - (3 / 2) x & sup2; = - 3 / 2 (x-1 / 3) & sup2; + 1 / 6
It is an increasing function on (0,1 / 3), and the function f (x) ＜ = 1 / 6
1 when n = 1, A1 ∈ (0,1 / 2)
A1

Mathematical induction proving senior high school proving problems
It is proved that 1 / (2 * 3) + 1 / (3 * 5) + 1 / (4 * 7) +. 1 / ((n + 1) * (2n + 1)) is less than 5 / 12

Strengthen to prove that 1 / (2 * 3) + 1 / (3 * 5) + 1 / (4 * 7) +. 1 / ((n + 1) * (2n + 1)) ≤ 5 / 12-1 / (2n + 2) n = 1, obviously if n = k-1 holds, n = k left ≤ 5 / 12-1 / (2k) + 1 / ((K + 1) * (2k + 1)) ≤ 5 / 12-1 / (2k) + 1 / ((K + 1) * (2k)) = 5 / 12-1 / (2k + 2)

The difficult problem of mathematical induction in college entrance examination
Given an = (1 + lgx) ^ n, BN = 1 + NLGX + n (n-1) / 2 (lgx) ^ 2, where n ∈ n, n > = 3, X ∈ (1 / 10, + ∞), try to compare the size of an and BN
Prove it by mathematical induction
BN = 1 + n * lgx + {[n (n-1)] / 2} * (lgx) ^ 2 well, it looks clearer. Thank you

When x = 1, an is equal to BN
When x ∈ (1 / 10,1), a3bn