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

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

A: Divergence of the series

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

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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 that 2n+1 > (n + 2) · sin(n) for all positive integers n.

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Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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Q: Prove by Strong Induction that, 2n-1+2n-2 = 3 x 2n-2 for every positive integer n.

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Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

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Q: -, first prove that )" 6.

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

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

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Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

A: Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns.  There is a relationship…

Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

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Q: Prove the formula () - (*: ) - (**) ---() -(*) r + n + 1 n +• r +1 by induction on n (for r…

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Q: Show that yn for all n=1, 2₁- ✓

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Q: 6. Prove that ER, # 0.

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Q: Let k be a positive integer. Prove ÷VE. Note: do not use induction. j=1

A: The solution are next step

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: 7. Prove that 2. O = (-1). %3D

A:

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 if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  =&gt; k | {(n+4) - (n-1)} =&gt; k | 5

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: о Prove 2 (2n – 1) = n n - п - 1 k=1

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Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

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Q: If nis a positive integer 1 then prove (D - a) n!

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Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: 1. Prove by Induction: 2 + 4 + 6+ .. + 2*n n * (n + 1) %3D for all positive numbers ofn>= 0

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

A: Answer

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: 10. Prove by mathematical induction that Σ k (k + 1) = n (n + 1)(n + 2)/ 3 1sksn

A: In induction method, the given hypothesis or formula P(K) is proved by establishing its validity for…

Q: se Euler's Theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 35. (You…

A: By using the fernet's theorem: If a and n are co prime. That is (a,n)=1 then aϕ(n)≡1modn For n=35…

Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: For n2 1. prove using mathematical induction: 2" s 2n+1 - 2n-1 - 1

A: We have to solve given problem:

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27