Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

Q: Use generalized induction to prove that n! < n^n for all integers n≥2.

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Q: 1), n = 1, 2, . .

A: In this question, the concept of mathematical induction is applied. Mathematical Induction Inductive…

Q: 16- Prove that for every positive integer n and all real number r except 1: -1 r -1 i=0 17- Show…

A: We need to prove the following given condition in question 16 and 17

Q: 3" 7.

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Q: (b) Use mathematical induction method to show that 품(주-)- (두-0) (주-)-(-)1 n+1 2n

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Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

Q: 1 Use Mathematical Induction to prove that 2 k2 k=1

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Q: 2) induction to that for all Use prove nonnes atine integers 1, 51 (n=-n)

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Q: prove  by mathematical induction. n! > n2, for each integer n  >= 4

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Q: Prove by induction, '2 divides n2+n whenever n is a positive intege

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Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: Prove that 3n<n! if n is an integer greater than 6

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: () is even. 3. Prove that for all positive integers n, the number

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Q: Prove by induction that the sum of the first n integers starting at 1 (1,2,3,...n) is n(n+1)/2.

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Q: -ん 2 kodd

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Q: Prove using the mathematical induction that for all integers n >= 1, 12 + 22+ . . . n2 = (n(n +…

A: The solution is given as

Q: Prove by induction that 1) 2 and 3 can divide n^3-n for all n ∈ N 2) n^2-7n+15 >=0 when n>=3 for…

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Q: Prove by induction that for any non-negative integer n > 4, 3^n < (n+1)!

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Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: Prove by mathematical induction that the sum of the first n even positive integers is n' +(2n) = n^2…

A: to prove 2+4+6+..2n=n2+n by induction  Prove the result is true for n=1 LHS  LHS = 2 RHS…

Q: Use induction to prove that for all integers n > 1. n Σ+1 = η n+1

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Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

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

Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: Prove that n2+n is even for all positive intrgers n 1.by using parity arguments (ie. concept of…

A: We have to prove that n2+n is even.

Q: Prove by mathematical induction that for all positive integers n 1: a 12 x 2 + 22 × 3 + 32 × 4 + ·…

A: Let P(n) be the given statement. Then P(n):12x2 +22x3 + ... +n2 (n+1)= (1/12)n(n+1)(n+2)(3n+1) Thus…

Q: For any n e N, 3 divides n – n. -

A: Prove using Mathematical of induction

Q: Prove using mathematical Induction: n 1 2. k=1

A: First we assume that result is true for n=k' then we prove that result is true for n=k'+1 .

Q: Prove using mathematical induction that for all n 2 1, n(3n – 1) 1+4+7+- .. .+(3n – 2) = 2

A: Mathematical Induction:  Let the statement to be proved be A(n), n≥1 Base case: We prove it for n=1…

Q: Prove that 2n>n2 if n is an integer greater than 4.

A: According to the given information, it is required to prove the given statement.

Q: 2. Prove by induction that for all integers n > 0 5n+1 - 1 1+5+52 + ... + 5" 4

A: prove of the above by induction is given below...

Q: Use generalized induction to prove -    1 + 2n < 2n2 for all integers n greater than or equal to…

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Q: Prove by mathematical induction that for all nonnegative integers n that…

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Q: 12 + 22 + ...+ n² = n(n+1)(2n + 1) 6

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Q: Prove by induction that 1 1 +...+ 3-4 1 for every n 21 n+1 1.2 2-3 n (n+1)

A: To prove the given statement.

Q: Use Mathematical Induction to prove that for all integers n > 1: (1 × 2) + (3 x 4) + (5 × 6) + · - -…

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Q: Prove the following equation for every n>=2 by mathematical induction.…

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Q: Prove by induction that (1+1/5)^n ≥ 1 + n/5 for all n∈N∪{0}.

A: Prove by induction that (1+1/5)^n ≥ 1 + n/5 for all n∈N∪{0}.

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: (1-) (1--)--(1-=) 1 n+1 2.

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Q: Prove by induction that for positive integers n: 1! x3! x5! x...(2n (2n-1):2 (n!)"

A: For n=1 Left hand side = 1!=1 Right hand side  =n!n=1!1=1 now 1≥1 The result is true for n=1 Assume…

Q: Prove that by mathematically Induction. 2^n < (n + 1)!, for all integers n ≥ 2.

A: We will prove the base case for n=2 Hence 22&lt;(2+1)!4&lt;3!4&lt;6 True Now assume the result is…

Q: Prove by mathematical induction that 2n =0

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Q: Prove that (1 -)(1-÷)(1-→).--(-) for all integers m 2 2 by m 2m nathematical induction.

A: *concept:- formula, a^2 - b^2 = (a+b)*(a-b).

Q: Prove by induction that for all n ∈ N with n ≥ 8, 2^n ≥ 3n^2 + 5

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Q: Prove that n! > n2 for every integer n ≥ 4.

A: Use mathematical induction, to prove the statement n!&gt;n2 is true for all integer n≥4. Substitute…

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.