Prove Γ (n) = (n-1)! where n ∈ ℕ

Q: 5. By Principle of Mathematical Induction, prove 2+ 22 + . 2" = 2n*1-2 for all n>1 %3D ...

A: Hi, thank you for the question. Since you have posted two questions, as per our policy we will…

Q: 2 If C = |2 -1 0 Show that (C")-1 = (C-')" -2 2 2

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Q: 2n+2 52n+1 3n+2, 22n+1 is divisible by 19 co dady M N [ day f04+ 204d (21)

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Q: 4. Use induction to prove that n 1 Σ(i)(i!) = (n+1)! - i=1

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Q: show that for all N in N * Σ (2/(n(n+2)=3/2 - 2N+3/(N+1)(N+2)

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Q: 6. Use the binomial theorem to prove that 3 = C(n,0) + 2C(n,1)+2²C(n, 2) + +2"C(n, n)

A: By using the binomial thorem to prove that, 3n=Cn, 0+2Cn, 1+22Cn, 2+...+2nCn, n

Q: k-1, O A. (n+1)! (k-1)!(n-k)! (n+1) В. (k-1)(n-k+2) OB. С. (n+1)! (k-1):(n-k+2)! D. (n+1)!…

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Q: f(n)={1 if n is odd; n if n is even  Prove or disprove. f(n) is O(n).

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Q: (a) How many different groups of 6 people could be selected from a group of 24 to go on a trip?…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: The factorial of a positive integer n, denoted n!, is equal to the product of all integers between 1…

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Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: B. Find the value of n: (n+3)t 1. (n+1)! = 20

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Q: For integers n>r> 0, use the definition of the factorial function to prove that (n − 1)! (n − 1)! n!…

A: By the definition of factorial function, the factorial of a non-negative integer n, denoted by n!,…

Q: Prove that, for n = 1, 2, ..., we have 1 1 + 1.2 2.3 +...+ n(n+1) = n n+1

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Q: Evaluate P(n, k – 1)P(n – 1,n – k). - k=1

A: We have to evaluate ∑k=1nPn,k-1Pn-1,n-k

Q: Let S= {n | n=5 k+ 4 for some integer k} Let T= {n|n=5 j– 1 for some integer j}

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Q: At a particular college there are a total of 75 teachers. The table below shows the number of…

A: There are total 75 teachers. We can select any 4 teachers from 75 teachers in 75C4=1215450 There are…

Q: Prove that µ(d) p(n) = n . P ulp

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Q: For any positive interger n, (n + 1)! / n! - n = (A) 0 (B) 1 (C) n (D) n + 1 (E) n!

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Q: P(n, r) C(n, r) None of these n! (n - r)! Or! ||

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Q: Prove that (2) (6) (10) (14) ...(4n-2) = (2n)! n!

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Q: {in(n² +1) Im(x +4)} 19. А. В. 1 С. 1/2 D. 2 E. Divergent

A: we have to identify the given sequence is convergent or divergent

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

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Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

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Q: Find all n such that σ(n) = 72.

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Q: Test { (-1,"! (n! (2m)!

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Q: P(13,5) for P(n,r)= n!/(n-r)!

A: find out P(13,5) using formula

Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: Prove that n2 < 2n for n in J, n ≥ 5

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Q: Show that for n ≥ 3, ô(n) is even.

A: We have from the definition of Euler ϕ function: ϕ(1)=1 and ϕ(2)=1 Now Let n ≥ 3

Q: 4n² +14n+1 = Let P(n):9+13+...+(4n+5) 2 Show P(k)→ P(k+1). Does this mean P(n) is true Vintegers n ≥…

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Q: Show that Σ dn o(n) n for every positive integer n.

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