Let P(n) be the statement that n(n + 1)(2n + 1) 12 +22 + .. +n? for n>0 6. What is P(1)? a. It doesn't exist b. n = 1 1(1+ 1)(2(1) + 1) 12 C. n(n + 1) (2n + 1) O d. 1 +22 + . +n? = 02 = 0(0+ 1) (2 (0) + 1) e. 6. ai

Q: 4.6.2. Prove the following inequalities by induction for all n Ee N. (a) 5" +5 < 5n+1 (b) 1+2+3+…

A: (a) The given statement is: 5n+5&lt;5n+1 We need to prove that the given statement is true for all…

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

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A: Consider the statement  ''For all odd primes p&lt;q there exist positive non primes r&lt;s such that…

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A:

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

A:

Q: ^n for all n>=6

A:

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A:

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: *3. Prove that 1² +2² + ... · +n² = n(n+1)(2n+1) for all n € N.

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

Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

A:

Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A:

Q: b) Prove the following by mathematical induction: n(n+1)(n+2) 1.2+2.3+3.4+..+n(n+1) : 3 for all…

A:

Q: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

Q: Argue by contradiction that for all integers n > 2, there is a prime number p such that n<p<n!,…

A: We have to prove that there is a prime number p such that n &lt; p &lt; n!, where n &gt; 2 and n! =…

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: 1 For all integers n21:+ 1.2 2-3 3. n(n +1) n+1

A: Calculate the left hand side of the given equation. 11·2=12 Calculate the right hand side of the…

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: (d) In an inductive proof that for every positive integer n, n(n+1)(2n + 1) 6 what must be proven in…

A:

Q: 9. Use mathematical lowing inequalities. induction to establish each of the fol- (a) [BB] 2">n², for…

A: As per policy, we are solving only the first Question, Please post multiple Questions separately.…

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: Prove that 13 +23 + 23 + .. + n3 = n²(n + 1)?

A:

Q: 3(3 + 1) n = 3: Show that 1 + 2 + 3 =

A: To show that 1 + 2 + 3 + ...  + n = n(n + 1)2 is true for the given value of n.   Also, to show…

Q: Use the permutation and combination formulas to prove the following. (a) For all integers n 2 2,…

A: We will solve the problem

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A: To prove: n4+n2+1n=Θn3

Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

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A:

Q: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

A:

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

A: We will solve the problem using recurrence relation

Q: Q4) Prove for any natural number n 20 Li 1+2+2+. +2" = 2-1 ミ (n+1) .... (11) 1+2+3+ + n = 2

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

Q: 9.59. a. Let (a][b] € Zs. If [a] · [b] = [0], does it follow that [a] = [0] or (b] = [0]? b. How is…

A: Prime number p has factor 1 and p

Q: Prove or disprove: 2 + 8 + 18 + ... + 2n^2 = n^2 + n

A: Please see the explanation below

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

A:

Q: What is wrong with the following proof that every positive integer equals the next larger positive…

A: Answer

Q: 7. What is the big O of each of the following expressions? a) (4N3 +6N2 + N) / 3 Big(O) b) 6log2N +…

A: (a). We have the expression 4N3+6N2+N3. Note that for a large value of N, the terms N and N2 can be…

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

A:

Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: - Let n and r be positive integers such that r≤n. Then (a) P = nx pn-1 (b) P = P-1

A:

Q: 04 Prove that m (n-k) = (n+1)- ("-m)n 2m n2 m 2

A: Note: Cr+1n+1=Crn+Crn-1+…+Crn-m+Crn-m-1+…+Crr→1 From identity 1, Crn-m=Crn-m-1+Crn-m-2+…+Crr Now,…

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

A:

Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A:

Q: only Q2(8) 2. Show that following statements are correct: į. 4n+100=0(n) iii. n³ ± O(n²) v. n! =…

A:

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

A:

Q: 9.59. a. Let (a][b] E Zs. If [a] · [b] = [0], does it follow that [a] = [0] or [b] = [0]? b. How is…

A: Solution by using definition of zero divisors:

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…

Q: If P(n): 2n < n! , nEN then P(n) is true for all n 2... None n23 n22 n24

A: If P(n):2n&lt;n!, n∈N then P(n) is true for all n≥...