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anemone
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Prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+2}}\right\rfloor$ for any positive integer $n$.
anemone said:Prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+2}}\right\rfloor$ for any positive integer $n$.
kaliprasad said:LHS = $\lfloor{\sqrt{n}+\sqrt{n+1}}\rfloor $
= $\lfloor\sqrt{(\sqrt{n}+\sqrt{n+1})^2}\rfloor $
= $\lfloor\sqrt{n+n+1+2\sqrt{n(n+1)}}\rfloor $
= $\lfloor\sqrt{2n+1+2\sqrt{n(n+1)}}\rfloor $
= $\lfloor\sqrt{2n+1+2\sqrt{(n+\dfrac{1}{2})^2 + \dfrac{3}{4}}}\rfloor $
= $\lfloor\sqrt{2n+1+\sqrt{(2n+1)^2 + 3}}\rfloor $
now realising that integral part of square root of x and square root of x + t where t is less than 1 are sameso we need to find the integral part of $\sqrt{(2n+1)^2 + 3}$
$(2n+1)^2 + 3\gt(2n+1)^2$
and $(2n+1)^2 + 3\lt(2n+2)^2$ as $(2n+2)^2-(2n+1)^2 = 4n + 3$so integral part of $\sqrt{(2n+1)^2 + 3}$ = (2n + 1)
so LHS = $\lfloor\sqrt{2n+1+2n+1}\rfloor $
= $\lfloor\sqrt{4n+2}\rfloor $
= RHS
The purpose of Floor Function Challenge V is to test one's understanding and application of the floor function in mathematics. It involves solving various mathematical problems using the floor function and its properties.
The floor function, denoted as ⌊x⌋, is a mathematical function that rounds down a given real number to the nearest integer. It returns the largest integer that is less than or equal to the input number.
The floor function is commonly used in computer programming to round down a number to an integer, which is useful for tasks such as truncating decimal places or generating random integers. It is also used in various mathematical proofs and calculations involving real numbers.
The ceiling function, denoted as ⌈x⌉, is similar to the floor function but rounds up a given real number to the nearest integer. This means that it returns the smallest integer that is greater than or equal to the input number. In other words, the ceiling of a number is the floor of that number plus one.
Yes, there are several special rules and properties of the floor function, such as: it can be applied to both positive and negative numbers, it always returns an integer, and it is discontinuous at integer values. It also follows the distributive, associative, and commutative properties, and has a relationship with the modulus function.