Sequence and mathematical induction 1. Known sequence a, a (1-A), a (1-A) 2 Is an equal ratio sequence, then the value range of the real number a is________ 2. "Three numbers a, B, C are equal proportion sequence" is the basic of "B2 = AC"________ condition 3. If three numbers x, 2x + 2 and 3x + 3 are in equal proportion sequence, then X=________ 4. In the equal ratio sequence {an}, a1 + A2 + a3 = 6, A4 + A5 + A6 = 48, find the general term formula of the sequence {an} 5. It is known that the sequence {an} is an equal difference sequence, and the sequence {BN} is an equal ratio sequence, and Sn = ean, TN = LG BN, where n belongs to n *. Verification: (1) the sequence {Sn} is an equal ratio sequence, and (2) the sequence {TN} is an equal difference sequence 6. If A4 = 5, a8 = 6, then A2 &; a10=________ ,a6=________ 7. It is known that {an} is an equal ratio sequence, and an ＞ 0, A2 A4 + 2 A3 A5 + A4 A6 = 25. Then the value of A3 + A5 is equal to________ 8. Known A1, A2 If the common ratio Q ≠ 1, then () A. A1 + A8 > A4 + A5 B. a1 + A8 < A4 + A5 C. a1 + A8 = A4 + A5 D. the size relationship between a1 + A8 and A4 + A5 cannot be determined 9. There are three numbers in equal proportion sequence, the product of which is 27, and the sum of squares of which is 91 10. Given that the sum of the first n terms of the sequence {an} is Sn = 3 + 2n, find the general term formula an

Sequence and mathematical induction 1. Known sequence a, a (1-A), a (1-A) 2 Is an equal ratio sequence, then the value range of the real number a is________ 2. "Three numbers a, B, C are equal proportion sequence" is the basic of "B2 = AC"________ condition 3. If three numbers x, 2x + 2 and 3x + 3 are in equal proportion sequence, then X=________ 4. In the equal ratio sequence {an}, a1 + A2 + a3 = 6, A4 + A5 + A6 = 48, find the general term formula of the sequence {an} 5. It is known that the sequence {an} is an equal difference sequence, and the sequence {BN} is an equal ratio sequence, and Sn = ean, TN = LG BN, where n belongs to n *. Verification: (1) the sequence {Sn} is an equal ratio sequence, and (2) the sequence {TN} is an equal difference sequence 6. If A4 = 5, a8 = 6, then A2 &; a10=________ ,a6=________ 7. It is known that {an} is an equal ratio sequence, and an ＞ 0, A2 A4 + 2 A3 A5 + A4 A6 = 25. Then the value of A3 + A5 is equal to________ 8. Known A1, A2 If the common ratio Q ≠ 1, then () A. A1 + A8 > A4 + A5 B. a1 + A8 < A4 + A5 C. a1 + A8 = A4 + A5 D. the size relationship between a1 + A8 and A4 + A5 cannot be determined 9. There are three numbers in equal proportion sequence, the product of which is 27, and the sum of squares of which is 91 10. Given that the sum of the first n terms of the sequence {an} is Sn = 3 + 2n, find the general term formula an

1. Known sequence a, a (1-A), a (1-A) 2 3. If three numbers x, 2x + 2, 3x + 3 are equal ratio series, then x = - 4.4

Proof of induction in Senior Two
Prove the inequality (1 + 1 / 3) (1 + 1 / 5) for all natural numbers n greater than 1 by mathematical induction （1+1/(2n-1)）>（√（2n+1））/2
I've searched one on the Internet. I think it's a problem, so don't copy it

1. N = 1, left = 1 + 1 = 2 > right 2. Suppose n = k becomes immediate (1 + 1 / 3) (1 + 1 / 5) (1 + 1 / (2k-1)) > (√ (2k + 1)) / 2 when n = + 1K (1 + 1 / 3) (1 + 1 / 5) (1 + 1 / (2k-1)) (1 + 1 / (2k + 1)) > [(√ (2k + 1)) / 2] (1 + 1 / (2k + 1)) the following only needs to prove [(√ (2

Proving inequality by mathematical induction
Prove the inequality 1 / N + 1 / (n + 1) + 1 / (n + 2) +... + 1 / (n ^ 2) > 1 (n belongs to N +, and N > 1)

(1) When n = 2, 1 / 2 + 1 / 3 + 1 / 4 = 13 / 12 > 1. So the inequality holds. (2) suppose that when n = k, 1 / K + 1 / (K + 1) + 1 / (K + 2) +... + 1 / (k ^ 2) > 1 holds. Then when n = K + 1, there is 1 / (K + 1) + 1 / (K + 1 + 1) + 1 / (K + 1 + 2) +... + 1 / ((K + 1) ^ 2) > 1 + 1 / (K ^ 2 + 1) + 1 / (k ^ 2 + 2) +