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原题呈现:23.(12分)有一列按一定顺序和规律排列的数:第一个数是1/(1×2);第二个数是1/(2×3);第三个数是1/(3×4);对任何正整数n,第n个数与第(n+1)个数的和等于2/(n×(n+2)).(1)经过探究,我们发现:1/(1×2)=1/1-1/2;1/(2×3)=1/2-1/3:1/(3×4)=1/3-1/4;设这列数的第5个数为a,那么a>1/5-1/6,a=1/5-1/6,a<1/5-1/6,哪个正确?请你直接写出正确的结论;(2)请你观察第1个数、第2个数、第3个数,猜想这列数的第n个数(即用正整数n表示第n个数),并且证明你的猜想满足“第n个数与第(n+1)个数的和等于2/(n×(n+2))”;
The original title shows: 23. (12 points) There is a list of numbers arranged in a certain order and regularity: the first number is 1 / (1 × 2); the second number is 1 / (2 × 3); the third For any positive integer n, the sum of the nth number and the (n + 1) th number is equal to 2 / (n × (n + 2)). (1) We find that 1 / (1 × 2) = 1 / 1-1 / 2; 1 / (2 × 3) = 1 / 2-1 / 3: 1 / (3 × 4) = 1 / 3-1 / 4 ; Suppose the number of the fifth column is a, then a> 1 / 5-1 / 6, a = 1 / 5-1 / 6, a <1 / 5-1 / 6, which one is correct? Write the correct conclusion; (2) Please observe the first number, the second number, the third number, guess the number of the n-th number of columns (that is, with a positive integer n that the number n), and Prove that your guess satisfies “The sum of the nth number and the (n + 1) th number equals 2 / (n × (n + 2))”;