**Problem 1:** Define a sequence as:

Prove that this sequence has a finite limit as Also determine the limit.

**Problem 2:**Let and be two sequences of numbers, and let be an integer greater than Define Prove that if the quadratic expressions do not have any real roots, then all the remaining polynomials also don’t have real roots.

**Problem 3:**Let be a cyclic quadrilateral with circumcentre and the pair of opposite sides not parallel with each other. Let and Denote, by the intersection of the angle bisectors of and and and Suppose that the four points are distinct.

(a) Show that the four points are concyclic. Find the centre of this circle, and denote it as

(b) Let Prove that are collinear.

**Problem 4:**Let be a natural number. There are boys and girls standing in a line, in any arbitrary order. A student will be eligible for receiving candies, if we can choose two students of opposite sex with standing on either side of in ways. Show that the total number of candies does not exceed

**Problem 5:**For a group of 5 girls, denoted as and boys. There are chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:

(a) Each chair has a proper seat.

(b) The order, from left to right, of the girls seating is

(c) Between and there are at least three boys.

(d) Between and there are at least one boy and most four boys.

How many such arrangements are possible?

**Problem 6:**Consider two odd natural numbers and where is a divisor of and is a divisor of Prove that and are the terms of the series of natural numbers defined by

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