For any positive integer n, the size relationship between 2N-1 and (n + 1) 2 is conjectured and proved

For any positive integer n, the size relationship between 2N-1 and (n + 1) 2 is conjectured and proved

When n = 1, 21-1 ＜ (1 + 1) 2, when n = 2, 22-1 = 2 ＜ (2 + 1) 2, when n = 3, 23-1 = 4 ＜ (3 + 1) 2, when n = 4, 24-1 ＜ (4 + 1) 2, when n = 5, 25-1 ＜ (5 + 1) 2, when n = 6   26-1 ＜ (6 + 1) 2, when n = 7   27-1 = (7 + 1) 2... (2 points) n = 8

If a is greater than 0 and N is a positive integer, try to guess the relationship between the n-th power of (- a) and the n-th power of A Let me give you a hint. N is less than 1 (not 0), n is equal to 1, and N is greater than 1

When n is an even number, two are equal; When n is an odd number, two numbers are opposite to each other

For any positive integer n, the size relationship between 2N-1 and (n + 1) 2 is conjectured and proved

When n = 1, 21-1 ＜ (1 + 1) 2,
When n = 2, 22-1 = 2 ＜ (2 + 1) 2,
When n = 3, 23-1 = 4 ＜ (3 + 1) 2,
When n = 4, 24-1 ＜ (4 + 1) 2,
When n = 5, 25-1 ＜ (5 + 1) 2,
When n = 6   26-1＜（6+1）2，
When n = 7   27-1 = (7 + 1) 2... (2 points)
When n = 8, 9, 10,..., 2N-1 ＞ (n + 1) 2,
When n ≥ 8, 2N-1 ＞ (n + 1) 2  … (4 points)
It is proved that: ① when n = 8, the above conclusion is tenable;
② Suppose that when n = K (k > 8), 2k-1 > (K + 1) 2,
Then when n = K + 1, 2 (K + 1) - 1 = 21 + (K + 1) = 2 • 2k-1 > 2 (K + 1) 2, and 2 (K + 1) 2 - (K + 2) 2 = k2-2, ∵ K ≥ 9 ∵ k2-2 > 0,
SO 2 (K + 1) 2 - (K + 2) 2 > 0,
That is, when 2 (K + 1) 2 > (K + 2) 2, that is, 2 (K + 1) - 1 > (K + 2) 2, that is, n = K + 1, the conclusion is valid,
It is known from ① and ② that the conclusion is valid for any n ≥ 8

After simplifying the n-th power ab of (- 1) and the N + 1st power ab of (- 1) (n is a positive integer), the result is

In the original formula, AB is extracted to obtain the 2n + 1 power of AB (- 1)
N is a positive integer, so 2n + 1 is an odd number
2n + 1 power of (- 1) = - 1
Original formula = - AB

Guess: what is the n power of (AB) when n is a positive integer? Try to prove the correctness of the conclusion How to prove it

（ab)^n=a^n*b^n
This is the property of power

If n is any positive integer, try to explain the N + 2 power of 3 - 4 × N + 1 power of 3 + 10 × The nth power of 3 can be divided by 7 Factorization

Original formula = 3 ^ n (3 ^ 2-4 * 3 + 10)
=3^n*7
Because 3 ^ n * 7 can be divided by 7
So [3 ^ (n + 2) - 4 * 3 ^ (n + 1) + 10 * 3 ^ n] can be divided by 7