10. Prove that (n) ≤ n/2 for n ≥ 8.

Let f(n) and g(n) be functions satisfying  f(n) = {sqrt (n), 1 + f(n+1)} if sqrt (n) is an integer otherwise and g(n) = {sqrt (n), 2 + g(n+2)} if sqrt (n) is an integer othersiwe for postiive integers n. Find the least positive integer n such that f(n)/g(n) = 4/7

how to solve these 2 questions about Discrete Mathematics??

4) Determine Σ (-1)" n! (x-5)" 1=0 34 the interval of conver gence for

6. Given that -뿔 (금) (규 R(x) 4! 1+. for x E (-, 3), where & is between x and 0, find an upper bound for |R|, valid for all x € [-,), that is independent of r and §. 21 2

14. Prove that for every positive integer n, E"- k24 = (n- 1)2"+1 +2. %3D 32. Prove that 3 divides n + 2n whenever n is a positive integer. 34. Prove that 6 divides n' integer. - n whenever n is a nonnegative

4. You arrive at a bus station where buses arrive every 15 minutes, but you don’t know when the next bus will arrive. Which is the appropriate notation to describe your wait time (X)? X~Uniform[0,15] X~AN(7.5,4.33) X~N(7.5,4.33) X~Exp(7.5)