Use mathematical induction to prove the following: (a) 1×3+2×4+3×5+...+n×(n+2)= "x(n+D(2n+7); for all natural numbers n. (b) n³< (n+1)²; n>2.

Q: (a) Suppose that n > 1 is an integer with prime factorization n = pf"p.p. Prove that (k, + 1)(k, +…

A: As both of them are different questions(not a subpart of other), so as per the guideline, I am…

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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Q: Find the first three terms of the expansion (4 – 2x)³.

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Q: (n+1)! (n-1)! = 10n

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Q: Show that P(n, 1) + P(n, 2) = (22) – 2 (2), for integers n ≥ 2.

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Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

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Q: The sum + is between 8 (A) and 3 (B) and 1 (C) 1 and 1 14/ D) 1 and 1/2 E) 1 and 2

A: The given is  Sum of this can be calculated  as           =                         =…

Q: the statement below true or false for n=1? ©George Mason University 2+ 4 + 6+ ... + 2n = n² + n + 1

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Q: 1+ x Σ Aa" (1 – 2)? n=0

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Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

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Q: Use induction to prove the summation formula п(n+ 1)(2n +1) 6 i=1 for all n E N. Hint: In inductive…

A: Use induction to.prove the following summation formula

Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

A: Using Euler theorem to prove it.

Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: Express the sum 3 + 5 + 7+ ... + (2k-1) using E-notation.

A: The given sum is 3+5+7+…+2k-1. Express the sum 3+5+7+…+2k-1 using ∑ notation as follows.

Q: Write the binomial expansion for (x+ 2y)*. (x + 2y)* =D

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Q: ) Find an integer having the remainders 3, 11, 15 when divided by 10, 13, 17, respec- tively.…

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Q: { :Find the first 3 terms for 2n-1 (2n-1)! (−1)n+¹

A: given sequence  -1n+1x2n-12n-1!

Q: . (a) Find the coefficient of r in the expansion of ‹ (² + ²)² (+ -)

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Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

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Q: If x is positive, the first negative term in the expansion of (1 + x)27/5 is (a) 5th term (b) 6th…

A: Introduction :

Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

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Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: 1.-Proof by mathematical induction that 5 + 9 + 13 + ··· + (4n + 1) = n(2n + 3) ...

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Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: Malik currently has 39 tulips in his yard. Each year he plants 3 more. (a) Which of the following is…

A: given data  malik havs 39 tulips in his yard each year he plants 3 more  so,  p0= 39 d = 3

Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

Q: Find the sum of 1/7 + 3/7 + 32/7 + 32 /7 + ⋅⋅⋅ + (3)n-1 / 7

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Q: If the sum 5+ 5(3)+5(3)² +...+5(9)* =5465 then the value of k is (a) 5 (b) 6 (c) 3 (d) 4

A: We know that 1+a+a2+a3+…+an-1=an-1a-1, if a≠1

Q: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

Q: Use mathematical induction to prove: 5+8+11++ (3n+2) - n(3n+7) 2

A: As mentioned we proceed via induction. The details are given as step 2.

Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: zu +16n + 20.

A: Given: An expression n2+16n+20 To find All possible remainder when it is divided by 11.As per our…

Q: (b) Show that 1 11 2 k(k + 2)(k + 4) 96 k=1

A: Given series is ∑k=1∞1kk+2k+4

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.