2. 1.2+2.3+3.4 + ... + n(n + 1) = n(n + 1)(n+ 2) 3. for any n e N 2+ 22 + 2+. 3 + 2n = 2n+1 -2 for anynsN 4.

Q: in is where n is an integer. equal to the above e(1+2n)i wheren is an integer. equal to the above…

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A: I have used only one formula of euler phi function

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A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: Use linearity and formulas attached to rewrite and evaluate the sums.

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Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

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Q: n If E (ak + bk) 2n and n+1 k=1 E(ak – br) +, then ak = n n+2 k=1 k=1 3 2 5 2 O None 2

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Q: For any n e N, let Bn = (n, 0). {Bn}=1 is descending. Show that N Bn = 4. n=1

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Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: n-1 For all n > 2, 4+ 3n4" – 4n+1 - j=1 9.

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Q: 9.To test a hypothesis involving proportions, both np and n(1-p) should a) Be at least 30 b) Be…

A: In general, ‘for testing the null hypothesis involving proportions np should be equal or greater…

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 3. Given n > 1, let S = E-1: Show that S, is not k=1 _k° an integer.

A: We are given that   Sn = ∑k=1n1/k   and n>1              ..(i) We have to show that Sn  is not an…

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

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Q: -), prove that for all n E N: (a) n-1 E(:)-E- n k k=1 k=0 (b) ¿(:)-- = 2" k k=0 (c) El-1)* E-1» (:)…

A: According to Bartleby policy we solve up to 3-subparts. Please repost #d and mention it.   b)…

Q: For n > 1 we have Σμ(α) d|n = [1] = Τ if n = 1 if n > 1.

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Q: :""n e Z*, if n³ = 3k for some k E Z, then n = 3q for some q E Z." %| %3D

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Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

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Q: Find L {(2t" – 3)° } . | O 4(2n)! 2n+1 12(n!) + sn+1 4(2n)! g2n+1 [ 2(n!) 8n+1 O 4(n²)! 12n! 9.…

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

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Q: Fix k, n E N with n > 2 and k <n – 2. Prove that: | )-(".) · ) (:) ) (^) = (":") 2 + k 2 n – 2 n n +…

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Q: 1. Prove that 12 + 22 + ...+ n² : n(n + 1)(2n + 1) %3D 6. for n e N.

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Q: For every n E N, 1+3+5+ ·.+ (2n – 1) = n² + 3.

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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: Find L {(2 - 3)}. 2(n!) 2 3 - sn+1 O 4(2n)! 12(n!) 9. - s2n+1 sn+1 O 4(n²)! sn2+1 12n! 9. - sn+1 O…

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Q: Show that for any n E N, n(n + 1)(2n + 1) k=0

A: The solution is given as

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

Q: 14. If n e Z, then 5n2 +3n + 7 is odd. (Try cases.)

A: By method of direct proof, prove the statement " If n∈ℤ, then 5n2+3n+7is odd"k∈ℤk∈ℤ as follows.

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: If (ar + bx) 8n and n+1 k=1 n Σ (α- br) 1 1. then ar n | n+2 k=1 k=1 O 2 None 9. 2 5 2 3 2

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Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

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Q: Let n e Z. n#-1,3,

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

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Q: 4n +14n+1 Let P(n):9+13+...+(4n+ 5) = 2 Show P(k) P(k +1). Is P(n) true Vintegers n21?

A: ANSWER:

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

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Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

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Q: Find L {(2 t" – 3) (t" + 2)} O 2(2n)! n! 82n+1 sn+1 2(2n)! n! - 82n+1 sn+1 O 4 n! 82n+1 n! n+1 O…

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Q: For the given statement Sn, write the three statements S1, Sk, and Sk+1.

A: Given, (a).  Sn=1+2+3+....+n=n(n+1)2(b). Sn=12+22+32+....+n2=n(n+1)(2n+1)6write the three statements…

Q: Encontrar el enésimo término de las sucesiones 1 2 3 4 4'9'16' 25 3n an 1 1 an (n + 1)! °n =+ (n +…

A: Consider the sequence  14,29,316,425,........

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

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Q: Find L {(2 t" – 3) (t + 2)} 2(2n)! s2n+1 sn+1 O 2 (2n)! $2n+1 n! sn+1 4 n! 82n+1 n! n+1 8. O 2(n²!)…

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Q: The number N of deer in a certain population t years after observation began is given by  N =…

A: Given: The number N of deer in a certain population t years is To find: The deer population after…

Q: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

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Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…