is equal to. k=1 n2 The sum O n+1 2n O n+1 O n? +1 2n2

Q: For any positive integer n, n(n+1)(2n+ 1) EP = 12 +22 + .. + r?

A:

Q: Which of the following expressions are equivalent to + 6n2 + 12n -? n 18n + 38 A 6n (n+2) 18 (n+2) C…

A:

Q: (-1)" n+1 nEN. E = %3D n

A:

Q: The sum of n,2n,3n terms of an AP are S1,S2,S3 respectively.Prove that S3=3(S2-S1)

A:

Q: solve 5.3 (using k(pi-k)/2) and setting k =pi/2 or by your own methods if you have easier ones

A: It is known to us that

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! %3D if n is non-negative integer and proof r-1 r>t

A:

Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: If Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10)

A:

Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: 3. Let n E Z be odd. Show that there exist two consecutive integers such that their sum is equal to…

A: Given n be a odd integer. Which is of the form n= 2k+1, k is an integer.

Q: d) 0.01n2 + 0.0000001n4 is O(n2) 0.01n + 0.0000001n* =0.0000001n for n>0, O(n*) for n>0 ,2(n°). is…

A:

Q: Which of the following statement is always true? (a) p(2n) = p(n) (b) p(n) is always even. (c) Edin…

A:

Q: Which of the following alternatives provides a direct proof to show that for all n eZ. If n- 1 is…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: C. n In n n=2

A:

Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

A:

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…

A: See the attachment

Q: Select all expressions that are equivalent to (2n +6)(n+ 3). O a 2(n? + 6n + 9) 2n+ 6n +18 Oc 2n…

A: Given,  => (2n + 6) (n + 3)  Taking '2' common from expression , => 2 (n +3) (n + 3)

Q: n IfΣ (a +b:) 2n and n+1 k=1 n (ar – br) = -, then Ea n+2 k-1 k=1 O 7 O None 5. 2 2 2

A:

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: in is where n is an integer. e(1+4n)i where n is an integer. O equal to the above equal to the above…

A:

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

A:

Q: MN || OP in O0 mMQ = 2(mNP) Given: Prove: M. 2.

A: Given:MN ∥OPProve:mMQ⏜=2(mNP⏜)

Q: If n is a positive integer, then (n + 2)! A = |(n + 1)! (n + 2)! (n+ 3)! is equal to (n+ 2)! (n +…

A: We need to find the determinant of the given matrix. To find the determinant we will use the…

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

A:

Q: Ex : Express the Sum o+ 2+4+6+8 in Sigma nata tion. -2

A:

Q: that Edn H(n/d)7(d) = 1 and Edn H(n/d)o(d) = n for all n, where r(n) = d(n) and σ(n) = σι (n) Σm d.…

A:

Q: Which expression is equal to E (2" + n²) k-1 n=2 O (2" + n?) +D ¬k-2 (2" + n2) n3D2 nk-2 O (2*-1 +…

A:

Q: n°(r + 4)" 2 .n20/3 infinite, type "infinity" or 'inr".

A:

Q: (-1)*-1 '(nIn n)? n=1

A:

Q: In? (n) +7+ + 2" lim n→∞ 2n + 5 +n121212 + 4n n2 (d) 4+ + In4(n) + 4" + (n + 4)!

A:

Q: Let S = { 1 - (-¹)" | n € N}. Determine inf S and sup S. Justify. n

A:

Q: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² + 7 is even}. Prove that A = B.

A: Given that              A=n∈ℤ ; n is odd   and   B=n∈ℤ; 3n2+7 is even

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

A:

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

A:

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: n(n – 1) is O(n²). 2 Show that

A: f(n) is O(g(n)) if there exist positive constants c and k such that f(n)≤c g(n) whenever n>k.…

Q: ъ) n EN COn n > 0, (2n)! < 22n . (п!)?

A:

Q: Given: 21 = 2 23 = L4 e || n Prove: 2 3

A: …

Q: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² +7 is even}. Prove that A = B.

A: 5.

Q: Show that the n-th tetrahedral number O_n is given by S_n= n(2n^2+1)/3

A: A tetrahedral number corresponds to a 3-Dfigure in the shape ofa tetrahedron made of spherical…

Q: Vn EZ, if n+5 is odd then n is even

A: This question is related to integer values. We can prove it by using direct or indirect way. Please…

Q: 5. (8x*y® + 3x³) dx + (12x°y³ + 3xy²) dy = 0

A:

Q: only Q2(8) 2. Show that following statements are correct: į. 4n+100=0(n) iii. n³ ± O(n²) v. n! =…

A:

Q: Let S = {n e Z: n> 2} and let ne S. If 2n +? < 5, then 4n2 + < 25. %3D Hint: Trivial or Vacuous…

A:

Q: nt 2 GIVEN: *In ( 1+) hai (-1)" (n +2) 2n+1(n + 1) 4. Compute the exact value of )` n=0

A:

Q: 1. For In 5n2 + 9n3 + 1, ɑn =n², prove or disprove: In O(an) (n → ∞)?

A: Given:  xn =5n2 + 9n3 + 1, αn=n2. we have to prove or disprove : xn =O(αn ) ( n→∞).

Q: a. b. In n {an} = {1 - (n In (n+1) 1 8n²16n +6 7=200 + T- en-

A:

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…

Q: If P(n): 2n < n! , nEN then P(n) is true for all n 2... None n23 n22 n24

A: If P(n):2n&lt;n!, n∈N then P(n) is true for all n≥...