Inequality 1 - Assignment

Pure Mathematics – Inequalities 1

p.1

Inequalities Assignment 1 1.

(a) Solve the inequality x − 6 ≤ 3 , where x is a real number.

(b) Using the result of (a), or otherwise, solve the inequality 1 − 2 y − 6 ≤ 3 , where y is a real number. 2.

Let n be a positive integer. (a) Let a > 0. (i)

If k is a positive integer, prove that a + a k ≤ 1 + a k +1 .

(

)

(ii) Prove that (1 + a ) ≤ 2 n −1 1 + a n . n

(b) Let x and y be positive real numbers. Using (a)(ii), or otherwise, prove that xn + yn ⎛x+ y⎞ ≤ . ⎜ ⎟ 2 ⎝ 2 ⎠ n

3.

(a) For any positive integer n, prove that t n − 1 ≥ n(t − 1) for all t > 0. (b) (i)

Let a, b and c be positive numbers. By putting n = 3 and t =

3

abc

in (a), prove that ab a+b+c 3 2⎛a+b ⎞ − abc ≥ ⎜ − ab ⎟ . 3 3⎝ 2 ⎠

(ii) Let y1, y2, …, yk+1 be positive real numbers, where k is a positive integer. Using (a), prove that y k +1 ≥ (k + 1)Gk +1 − kGk , where Gk = k y1 y 2 K y k and Gk +1 = k +1 y1 y 2 K y k +1 . (iii) Using mathematical induction and (b)(ii), prove that x1 + x 2 + L + x n n ≥ x1 x 2 L x n n

For any n positive real numbers x1, x2, …, xn.

Pure Mathematics – Inequalities 1

4.

p.2

(a) Let a and b be non-negative real numbers. Prove that

(a + b )n ≥ a n + na n−1b

for all n = 2, 3, 4, ….

Write down a necessary and sufficient condition for the equality to hold. (b) Let {a1, a2, a3, …} be a sequence of positive real numbers satisfying a1 ≤ a 2 ≤ a3 ≤ L . For any positive integer n, define 1 a1 + a 2 + a 3 + L + a n and Gn = (a1 a 2 a 3 L a n ) n . n Prove that Ak +1 ≥ Ak for all k = 1, 2, 3, ….

An =

(i)

(ii) Using (a), prove that Ak +1

k +1

≥ Ak a k +1 for all k = 1, 2, 3, …. k

Hence prove that An ≥ Gn and An = Gn if and only if a1 = a2 = a3 = … = an for all n = 1, 2, 3, …. (c) Let n be a positive integer. Using (b), prove that n

n + 2 ⎛ n + 1 ⎞ n +1 >⎜ ⎟ . n +1 ⎝ n ⎠

1 ⎞ ⎛ Hence deduce that ⎜1 + ⎟ ⎝ n +1⎠

n +1

n

⎛ 1⎞ > ⎜1 + ⎟ . ⎝ n⎠