(a) By combinatorial reasoning, prove that for any n ≥ k ≥ 0 we have (^) + (* + ¹) + - + (*) = (x + 1) k [Hint: for each subset SC {1, 2,..., n + 1}, |S| = k + 1, consider the value of the largest element of S.]

Q: 2 4) Prove that n + 5 > 4 n for all integers, n >0.

A:

Q: Find a Big-O estimate of (nn+n2n+3n)(n!+3n)

A: Consider the function fn=nn+n2n+3nn!+3n The objective is to give big-O estimate for the given…

Q: 13. Floor and ceiling notation were defined in Section 1.1. (a) Show that []+[=n for all positive…

A: Floor and ceiling.

Q: By combinatorial reasoning, prove that for any n ≥ k ≥ 0 we have k+1 n+ ()+(*+¹) ++ ( )= (+1) k…

A:

Q: If Z ~ N(0,1), compute P(Z2 < 3)

A:

Q: 10. Show that the interval (0, 1) is covered by the open intervals In (1/n, 1- 1/n), n 2 3, but (0,…

A:

Q: Prove that for all integers m>0 , m^2+5m+4 is composite.

A:

Q: Show that z / [(z-1)(z-3)] = -3 Σ [(z-1)n / 2n+2] - [1 / (2(z-1))] for 0 < |z - 1| < 2.

A: by using techniques of partial fractions and series expansion of 1/(1-x)

Q: Construct a truth table for the following Boolean function showing all columns: (x + y')(x' + z')(y'…

A: 1'=0 0'=1 1+1=1+0=0+1=1•1=1 0+0=1•0=0•1=0•0=0

Q: k It is known that ( + (₁ ²₁₂) + (k) = (²+¹) (k + 1). Prove that r - 2 (k) + ( k + ¹ ) + ( k + ²) +…

A: We will use given formula and value of nr . nr=n!r!×n-r!

Q: 4. The conventional algorithm for evaluating a polynomial anx" + an-1x¹ +... + a1x + ao at x = c can…

A: Algorithm Polynomial (c, ao, a1,...,an: real numbers) power = 1; y = a0 ; for i = 1 to n power =…

Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

A:

Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

A:

Q: X = factorial(3)   def factorial(n): if n == 1: return 1 else: return n* factorial (n-1) def…

A: Definition of Mathematical Induction:       It is essentially used to prove that a statement P(n)…

Q: . You are traveling on a grid from (2, 2) to (18, 18). The only steps you can make are either to…

A: Solution of the question is given below.

Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: 5. Using the limit method, show that every polynomial p(n) = a¿n² + ak- belongs to O(n*). ... + ao,…

A: Big theta notation is an order notation. It determines the growth rate of two growth functions. It…

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A:

Q: 6. Suppose that 2 = 1, 2, 3, ... and that P(k) = p(1 - p)k-¹, ken. Let A be the odd numbers in 22.…

A: Note :As per guidelines only first 3 subparts have been answered if you need answers for the rest of…

Q: Suppose a, ne Z and 0 < a <n. Prove that n (2) = (^=¹1)+(21) a a-1 n- in two different ways: firstly…

A: