4. Determine whether or not the following is true: n+1 (a) i2i = n. 2n+² + 2, Vn ∈ N, n>1. (b) i=1 6[n(n² + 5), \n € N, n ≥ 1.

Q: n(n+1)(2n+1) Prove: 12 + 22 + 3² + ... +n? for all positive integers, n.

A:

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

A:

Q: (10) Find Std(X) when X is a discrete random variable with the following given density. (i) f(z) =…

A: Solution Dear student  as  per guidelines we answer the first question only when multiple different…

Q: 6. Show that, for integer s, S. s! | x25+lexp(-ax²)dx =: 2a +1

A: Note: Given proof statement has a little mistake, I have proof the actual result. First i have…

Q: 133. -Σ 2" Sn+ -(x + 2)" n=0

A: Consider the series, Rewrite this as Now take the constant outside,

Q: Find the integral

A:

Q: Σ1 (−1) = (n+1)(-1)", 2

A: We have to prove  Prove that ∑ni=1 i2(−1)i = ((n+1)(−1)nn)/2 , where n is a positive integer. We…

Q: 4. Conclude that n2 n=1 ||

A:

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Suppose that q 1 is a constant. Calculate the sum n Sn = kqk = q + 2q² + 3q³ +…+nq” k=1

A:

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: Show that |Z(Dn)|=1 or 2 according to whether n is odd or even

A:

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

A:

Q: Prove that : L²₁ (1-x²) Pm Phd x T 0. 2n (n + 1) 2n + 1 4 mn. m = 11. m, ne N

A:

Q: 28. Show {Xn, n ≥ 1} are independent if o(X₁,..., Xn-1) || σ(Xn) are independent for each n ≥ 2.

A: Given::- Show (Xn, n ≥ 1} are independent if o (X1,..., Xn-1) || σ (Xn) are independent for each n ≥…

Q: Show that for all n ≥ 1, the n ₁²>(n+¹) " +1\ in

A: We have to show that for all n⩾1 , the inequality                  ∏j=1njj⩾n+12n(n+1)2Since…

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: Let y1, . ... Yn real numbers, and let n y = 1/√ Σ² ²=1 Yi n their arithmetic mean. Show that for…

A: The proof is explained in the explanation section.Explanation:Solution: Let's prove the given…

Q: Below is an incomplete proof Prove that 3n2 > 2(n+1)2 for n ≥ 5 3n2 > 2(n+1)2 Is equivalent to 3n2 >…

A: The question already shows how the proof for 3n2 > 2(n+1)2 for n≥5 is equivalent to showing n2…

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: Show that n 1 i2 i=1 <2 - n

A:

Q: (1.2) Show that m m+n+1] m k=0 for any two positive integers m and n.

A: Please check the next step for detailed answer.

Q: Let n 2 2. Assume that Fn is prime. show that -1.

A:

Q: 29. For any odd integer n, n° n² +3 %3D 4 4

A:

Q: (15 – b) (9 – b) - z = 6. 6. För a = 1 och b + 3 saknar systemet lösningar. För a = 1 eller (4t) , y…

A:

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: 2F. n 1 ;;1=2" (b), (26)

A:

Q: n+1 Ei.2¹=n2"+2 +2 i=1 For all integer n >= 0

A:

Q: 1. Compute the value of (), for n > 3.

A: Solution

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

A:

Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: 6. Let A = {nk Z, n = 6k+5} and B = {nn is odd}. Prove that AC B.

A:

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: Q9) In University of California among the placement cell, P(students getting placed )=2n/9,…

A: P(students getting placed)=P(students not getting placed)=

Q: Σ s2 an convergent or divergent?

A:

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

A:

Q: Prove that n Sk k=1 = n(n + 1)(2n + 1) 6 for all n € N.

A: To prove:We will use mathematical induction to prove this.Let

Q: 5. For n ≥ 1, establish the formula (2n + 1)² = (4tn + 1)² – (4tn)².

A: