- Give an example of a sequence of positive rationals {a„}, such that the following statement holds: Vr, y so that 0 1 such that x < an < y. n=1

(3) 3 (b) Express x +1 1 as a single fraction in its simplest form. DO NOT WRITE IN THIJI

(2) Explain why the rational numbers are those having periodic representation or with- out infinite expansion.

Let n > 3 and 2 (;) ')<(3) k n – k + + in two ways: (a) algebraic proof, and (b) combinatorial proof Hint for (b): show that the left side counts only some of the things counted by the right side.

6. (i) Sketch the proof of the existence of a positive number satisfying the equation x2 = 5. We denote this positive number by the symbol V5. (ii) Prove that the number V5 is not a rational number (that is, it cannot be expressed as a quotient of integers).

Prove by legendra polynomial

True or false, for any integer x, x^2, - x will always result in an even number