# Difference between revisions of "1978 AHSME Problems/Problem 18"

(Created page with "==Problem==") |
Cleverbryce (talk | contribs) (→Solution) |
||

(One intermediate revision by one other user not shown) | |||

Line 1: | Line 1: | ||

==Problem== | ==Problem== | ||

+ | What is the smallest positive integer <math>n</math> such that <math>\sqrt{n}-\sqrt{n-1}<.01</math>? | ||

+ | |||

+ | <math>\textbf{(A) }2499\qquad \textbf{(B) }2500\qquad \textbf{(C) }2501\qquad \textbf{(D) }10,000\qquad \textbf{(E) }\text{There is no such integer}</math> | ||

+ | |||

+ | ==Solution== | ||

+ | Adding <math>\sqrt{n - 1}</math> to both sides, we get | ||

+ | <cmath>\sqrt{n} < \sqrt{n - 1} + 0.01.</cmath> | ||

+ | Squaring both sides, we get | ||

+ | <cmath>n < n - 1 + 0.02 \sqrt{n - 1} + 0.0001,</cmath> | ||

+ | which simplifies to | ||

+ | <cmath>0.9999 < 0.02 \sqrt{n - 1},</cmath> | ||

+ | or | ||

+ | <cmath>\sqrt{n - 1} > 49.995.</cmath> | ||

+ | Squaring both sides again, we get | ||

+ | <cmath>n - 1 > 2499.500025,</cmath> | ||

+ | so <math>n > 2500.500025</math>. The smallest positive integer <math>n</math> that satisfies this inequality is <math>\boxed{2501}</math>. |

## Latest revision as of 22:17, 30 September 2021

## Problem

What is the smallest positive integer such that ?

## Solution

Adding to both sides, we get Squaring both sides, we get which simplifies to or Squaring both sides again, we get so . The smallest positive integer that satisfies this inequality is .