Q) Let be three natural numbers such that and . Suppose there exists an integer such that form the sides of a right-angled triangle. Prove that there exist integers such that Before we go to the solution of this problem, let us get into some theory of Pythagorean Triplets. We define a triplet of integers as a Pythagorean triplet, if form… Continue reading
Week -8 | Inequalities – 2
1.1 Convex Functions and Jenson’s Inequality A function is called a convex function if the line segment between any two points on the graph of the function lies above or on the graph. The following curve illustrates this oncept of a convex function and concave function: Fig 1.1 For a convex function , we… Continue reading
Week -8 | Inequalities – 1
1.1 Introduction We have already introduced the concept of inequations in our ‘Theory of Equations’ modules. To recollect: and with both this implies with this implies . We shall introduce ourselves to some advanced inequalities that are used in the modern world. 2.1 The AM-GM-HM Inequality Consider the positive real numbers . The following inequality is called… Continue reading
Week -7 | Sequences & Series – 9
This is an optional module for those preparing for Engineering examinations Q-1) Do there exist distinct positive integers such that form an arithmetic progression (in some order)? Q-2) Define a sequence by , and for . a) Show that for and , divides b) If divides , show that divides . Q-3) Consider a nonconstant arithmetic progression . Suppose there exist relatively… Continue reading
Week -7 | Sequences & Series – 8
Practice problems MCQ questions can have more than 1 correct answer. Q-1) If are in HP, then are in a) A.P b) G.P c) H.P d) None of these Q-2) Find the value of Q-3) The value of is a) b) c) d) None of these … Continue reading