Show that the square roots of two successive natural numbers greater than `N^2` differ by less than `1/(2N)dot`

The roots alpha and beta of a quadratic equation are the square of two consecutive natural numbers. The geometric mean of the two roots is 1 greater than the difference of the roots. If alpha and beta lies between the roots of x^2-cx+ 1 = 0 then find the minimum integral value of c.

The roots alpha and beta of a quadratic equation are the square of two consecutive natural numbers.The geometric mean of the two roots. is 1 greater than the difference of the roots.If alpha and beta lies between the roots of x^(2)-cx+1=0 then find the minimum integral value of c.

The square root of a positive number less than 100 lies between:

What would be the digit in the 10th place of 7^n where 'n' is a natural number greater than 1 ?

If n is an odd number greater than 1, then n(n^(2)-1) is

Let n be the smallest nonprime number greater than 1 with no prime factor less than 10. Then

A number is greater than 3 but less than 9 also it is greater than 6 but less than 9.

If two numbers are selected at random from the set of first 20 natural numbers, then the probability of getting a number less than 13 and the other greater than 17 is

Show that the arithmetic mean of the square root of x cannot be greater than the square root of its arithmetic mean.