Question

Determine the number of positive integers n that satisfy 1/2 < n/(n +1) < 99/101. Do any negative integers satisfy this inequality? Hint: Tackle 1/2 < n/(n + 1) and n/(n +1) < 99/101 separately.

Answer 1

No

Step-by-step :

n/n + 1 > 1/2, (n + 1)/n < 2, (n+1)/n - 2 < 0, [(n + 1) - 2n]/n < 0, (1 - n)/n < 0, n > 0, 1 - n < 0, n > 1

n/n + 1 < 99/101, (n + 1)/n > 101/99, [99(n+1)]/n > 101, [(99n + 99)]/n > 101, [(99n + 99)]/n - 101 > 0, (99n + 99 - 101n)/n > 0

(99 - 2n)/n > 0, n > 0, 99 - 2n > 0, 2n < 99, n < 99/2

1 < n < 49.5