The known sequence {an} satisfies the following conditions: A1 = 1, a (n + 1) = 2An + 1, n ∈ n * (3) proof: n / 2-1 / 3

The known sequence {an} satisfies the following conditions: A1 = 1, a (n + 1) = 2An + 1, n ∈ n * (3) proof: n / 2-1 / 3

(3) First of all, it is better to prove on the right that an / a (n + 1) = (2 ^ n-1) / (2 ^ (n + 1) - 1)

You can't do an = 2 ^ n + 1 to prove A1 / A2 + A2 / A3 +... + an / an + 1 > n / 2-1 / 3

Suppose that when n = k, A1 / A2 + A2 / A3 +... + AK / AK + 1 > k / 2-1 / 3 is also true, then when n = K + 1, the left side = A1 / A2 + A2 / A3 +... + AK / AK + 1 + (AK + 1) / (AK + 2). > k / 2-1 / 3 + (AK + 1) / (AK + 2)

The known sequence an satisfies A1 = 1, an = 3 ^ (n-1) + a (n-1) (n = > 2). (1) find A2, A3; (2) prove an (3 ^ n-1) / 2

When n = > 2
An = 3 ^ (n-1) + an-1, namely an-an-1 = 3 ^ (n-1) an-1-an-2 = 3 ^ (n-2). A4-a3 = 3 ^ 3, a3-a2 = 3 ^ 2
Add the above formula, left = an-a2, right = 3 ^ 2 + 3 ^ 3 + +3 ^ (n-2) + 3 ^ (n-1) is the sum of equal ratio sequence
If A2 = 4 is obtained from a2-a1 = 3, then an = 4-9 [1-3 ^ (n-2)] / 2 = (3 ^ n-1) / 2
One of the answers to the sequence of equal ratio and equal difference