Part A Prove the following theorem using either regular or strong induction. Theorem: Suppose that n ≥ 2 people are at a gathering. Every person shakes hands with every other person, but not themselves. Then, the number of total handshakes which occur is n(n-1)

Q: Let n be a positive odd integer number (i.e., n = 2k + 1, for some integer k). Find a unit fraction…

A: To find: Unit fraction representation of 3n, for any positive odd integer n.

Q: Prove the Triangle Inequality |v+w| <= |v| + |w|  (Hint: Use Cauchy Schwartz)

A:

Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

A:

Q: solve using heaviside expansion theorem

A: To find: Inverse Laplace transform of 3p+1p-1p2+1 using Heaviside expansion theorem.

Q: Consider a population that grows according to the recursive rule ��=��-1+25, with initial population…

A: The objective of the question is to find the population at different stages using the given…

Q: Find Im(z) if z(8 + i) = 7+ (384 + 448i)

A:

Q: What is the cocfficient of x" in the power series expansion of v1 – 5x?

A: Given,

Q: Compute ATBT and BT AT by hand if possible. If not possible, describe why. A = 2 3 −1 0 B = 10 -1 0…

A:

Q: 1/sqrt(1) + 1/sqrt(2) + . . . + 1/sqrt(n) = sqrt (n) for n=1,2,3

A: We have to prove  11+12+13+...+1n≥n For n=1 11=1 For n=2   Suppose for n=k the statement is true…

Q: Prove that for every positive integer 1.2' + 2· 22 + 3 - 2° + ... +n · 2" = (n – 1)2"+1 + 2.

A:

Q: Prove that : Σ j=1 m=1 2.

A:

Q: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: R² = gen gen

A: see below the reason

Q: Let p be a polynomial of degree n and let E = { x | e^x=p(x)}.   Prove that the number of elements…

A: Given:   E = { x | e^x=p(x)}         E ≤ n+1

Q: Consider a population that grows according to the recursive rule P = Pn-1 + 110, with initial…

A:

Q: c) Show that for 2" > 4n for n > 5.

A:

Q: 6. Let c1, C2, C3, ... be the sequence defined by the following recurrence relation: C1 = C2 = C3 =…

A:

Q: r's = sr, for all 0 <i <n. [Proceed by induction on i and use the fact that ri+'s =r(r's) together…

A: The given statement is : ris=sr-i ,∀0≤i≤n

Q: 2 Vi – 1 - ) dt

A:

Q: 4. Let a 1. 2 Follow these steps to show that (x„)n€N is Cauchy. (a) Draw a picture and let L = b –…

A: Given - Let a &lt; b. Let x1 = a, x2 = b, and xn+2 = xn+1 + xn2 for n ≥ 1. To Show - Use induction…

Q: Part B Here are three attempted inductive proofs of the correct theorem. None of them are correct.…

A: "Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Use induction to show that: 3º + 3¹ + 3² + . . . . . + 3″ 1 = 1 2 (3' n+ - 1) This is a math proof…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

A:

Q: 2p 4-1(e) + Joi(e) = 空p(e). %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…