7. Prove that if a, = 1, a1 = 4, and a, = 6a,-1 – 9an-2 for n > 2, then a, = 3"-(3 + n) for all n 2 0. Show all your algebraic steps! Basic case(s): Inductive Hypothesis: Inductive step - we need to prove:

Q: 2. Let N = {A, | ne N}, where A, = {n,n+ 1, n +2, ...}. (a) Find UN and prove your conjecture. (b)…

A: we have given Ω = An n∈N where An =n, n+1, n+2,...... we have to find (a) find ∪Ω and prove your…

Q: n(n+1)(2n+1) 2.Prove k2 = for n > 1. 6. k=1

A:

Q: 2. Find an explicit formula for аn 3 За,-1+ 10ар-2 and ao %3D 0, ај 3D 7 An where

A: The solution is given as

Q: 2. (Proof by Contradiction) For every n E Z, 4 t (n² – 3).

A:

Q: 13 e, thank you.

A: e)Given that, sum of the first n terms of the given series is

Q: 2 2/The third term of a geometric sequace whose First term 2 and its FiFth tom 162 is ? A- az a 18…

A:

Q: 2. Let n be a positive integer. Show that 4n 4n-k (4n 3 4n k=2n (Hint: an argument similar to the…

A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

A:

Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: If (n + 1)(n + 2)an+1x" - nan-1x" = 0, n=1 show that for k = 1, 2, 3, ..., we have 1.3.5... (2k - 1)…

A:

Q: * :The nth term formula for { 1,0,1,0,... } is 1-(-1)" 2. an = A₂ = An خيار 2 1 + (−1)" 2.

A:

Q: to: explicitly state the te- Σ2/ n=1

A:

Q: 41. If yo Yı • Y2 , Y6 are the consecutive terms of a seies, ..., then prove that Yz = 0- 05 (yo+…

A:

Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

A:

Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A:

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

A:

Q: 3. Prove that for any positive integer n, n is even if and only if 5n + 2 is even. Indicate which…

A:

Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

Q: Suppose that Z~N(0,1). Then find Pr(2.78 < Z < 2.84)

A: Given data, Z~N(0,1) That is Mean=0 Standard deviation=1 P(2.78 &lt; Z &lt; 2.84)=?

Q: Now we assume that P(r) is true, that is, r can be written as a sum of 3s and 5s, for any cases,…

A: We are given a statement and its proof. We need to answer two questions about the proof. More…

Q: 2. Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n. for n ≥ 3, an = 2an-1+ bn-1…

A: NOTE:- AS POSTED MULTIPLE INDEPENDENT QUESTIONS. WE ARE ANSWERING ONLY FIRST QUESTIONS FOR YOU AS…

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

Q: Consider that: 1 + 2 + ... + n = An^2 + Bn + C  For all n and for some constant A, B and C.…

A:

Q: 13. Prove E-1(4i – 3) = n(2n – 1) for natural numbers n > 1 %3D

A:

Q: Prove n (2n = n n 1 k=1

A:

Q: (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of n=0

A: The ratio test can be used to assess if a series is converging or diverging. The following are the…

Q: 8. How many integer solutions are there to the inequality Y1 + y2 + Y3+ y4 0, 0 < y2 ≤ 10, 0≤ Y3 ≤…

A:

Q: 2. Prove that n²>2n for n >2

A: Note that, the given statement P(n): n2&gt;2n for n≥2 has little correction because if n=2  then…

Q: 8. If n+1, 2n-1 and n+5 are in AP. Find the value of n

A:

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

A:

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: 14. Find all solutions to 2sin(5x)-3V3 =-4/3. State exact values only. %3D

A: Given equation, 2sin5x-33=-43

Q: The sum of n,2n,3n terms of an AP are S1,S2,S3 respectively ,Prove that S3=3(S2-S1)

A:

Q: Q2/ prove [A] [A] = [I] if [2 4 -5 [A]= 2 7 1 3 4

A:

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

A:

Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: (-1)" (n+ 2) 2n+1(n + 1) Compute the exact value of

A:

Q: 13 + 33 + 53 + + (2n – 1)3 = n²(2n² – 1) - ... Ensure that you explain the steps taken.

A:

Q: at is wrong with the following equatlon: 8 + (6/2) = 17 -5 %3D A The symbology is not standard so…

A:

Q: 4 (a) Write out the multiplication table for Z6. How many times does [0]6 appear in each row of the…

A:

Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

A:

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

A:

Q: 4.By applying the Convolution Theorem to calculate L-1 is obtained: Determine the value of (a+b): a…

A: We have to find the solution.

Q: 4 and for n > 3, an 2an-1 + an-2 + 5n, express an in terms of n %3D 2. Let, a1 = 3, a2

A:

Q: 8. Working over R, suppose that M = PDP ' where P 1 Then, for any positive integer k, we have that…

A: To find Mk =?

Q: (16) Show that AB=BA for 1 5 0 0 A = 0 3 0 0 0 and B = 3 1 2

A:

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…