3 for all n.

Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

Q: Prove that if n is a multiple of 4, then Fn is a multiple of 3.  

A:

Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: n = n and n = 1. n Show that in

A: A factorial is the product of consecutive integers starting from 1. It is known that nr represents…

Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: n! diverges because an > 0. for all n > 1. True False

A:

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

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Q: Show that n!> n3 if n is large enough.

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: 2 3 n a. If n E N, then + + + •+ 3! 4! (n+1)! (Hint: 3! (reads 3 factorial) = 3 x 2 x 1. n! = nx…

A: Induction theory: Let the given statement is P(n). Step-1: We need to check P(n) is true for n=1.(or…

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

A:

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

A:

Q: nd the smallest positive integer N such that n+2 3n-4 -}| < < 0.01 for all n ≥ N.

A: We want to find N such that n+23n-1 - 13 &lt; 0.01    for all   n ≥ N

Q: п For all integers n > 1, 1.2 2.3 п(п+ 1) п+ 1

A:

Q: Σ (-1) n=1 nyerges for all n20

A: Please refer to the image below

Q: For all integers n> 1, 2+4 + 6+ . ..+ 2n = n² +n.

A: Here we are going to solve only the first question and for the remaining question please post the…

Q: If n>1 prove the sum %3D is not an integer. 1无

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: 1 + 2n < n° for all integers n² 2

A:

Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: all n N we have We apply this successively for n = bn+1 bn N, N+1, n 9 bN+1 < (m + €)bN bN+2 < (m+…

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Q: Find all n such that σ(n) = 72.

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Q: 1 If n is a positive integer, then (D-a %3D n!

A:

Q: Show that for any r, n > 1, п! Σ = r". nį!n2!...n,! nį+n2+…+n,=n, n¡ 20 11 (

A: Given information: The sample size is n. The number of categories is r.

Q: Find the largest value of n such that: Σ E (9k – 5) < 3000 k=1 n =

A: We expand the sum and use the know formulas of "sum of first n integers"

Q: 4 + In(n*) \ D) { { } [ Select ] 2 + In n 4 + In(n*) E) n + In n [ Select ] 2n? + 2n + 4 F) { (-1)".…

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Q: For all integers n> 2, 2" < (n+ 1)!.

A: In proof by mathematical induction, we assume that this is true for n=k. Then, we proof that the…

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

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

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Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

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Q: Compute the following: N!(N-4)!

A: Note that n!=n×n-1×n-2×…×3×2×1. Consider the given expression N!N-4!. This can be written as…

Q: Σ >k· k! = (n + 1)! – 1 k=1 for every positive integer, n

A: We will prove this by principle of mathematical induction.

Q: 1. Prove that if n e N, then E-1 k(k + 1) = n(n+1)(n+2)_ k=1 3

A:

Q: Show that...

A: We have to show that: n1=nn-1 for n≥1. We know the formula, nr=Crn=n !r ! n-r ! Applying above…

Q: 4. Prove that for every n > 1,3 divides n³ – n.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: teger n is odd if and only if 3n + 1 is even.

A: Here I am providing my handwritten solution

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

A:

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

Q: k+1 In k + 2

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Q: (b). For all positive integers n 2 2. (") = 2 (5) + n².

A:

Q: Find all positive integers n such that p(n)|n.

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Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

A:

Q: Find σ(n) if n = 2^k-1 (2^k - 3) where 2^k - 3 is prime

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Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

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Q: -6an-1 for n > 2 with the initial condition a = -9. an an

A: Given an = -6an-1

Q: Use induction to prove the inequality n< 2" for all n E N.

A: The objective is to prove the inequality.