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: 10. Give a combinatorial proof: For n ≥ 1 and k ≥ 1, (n)k = n · (n − 1)k-1.

A: To give: A combinatorial proof: For n≥1 and k≥1, k·nk=n·n-1k-1.

Q: HW 2: If Z - N(0,1), find a) p(Z > 1.2) b) p(-2 <Z< 2) c) p(-2 <Z < 1)

A: If Z~ N(0,1)   which means clearly it is standardized normal distribution table.

Q: 27 Find (all) the value(s) of c such that 3c" n= (A) c=3 (В) с— 4 (C) c = - (D) c=} 33 %3D (E) с 3…

A:

Q: let X1, X2, ..... XN Find UMVUE for 0² N(0,1) 0 € (-∞0,00)

A: Dear student, If you find this solution helpful please upvote ? it.

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: Evaluate: [- e2x dx J ex+1

A: SOLUTION-

Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: п(п + 1) 15. Verify that E-1k is equal to for the cases (a) 2 n = 2, (b) n = 8, and (c) n = 9.

A: We have to verify that ∑kk=1n=n(n+1)2 for the cases (a) n=2 (b) n=8 and (c) n=9

Q: Show that if (B) (C) N N then the value of neither x nor y can exceed 1/4. (A) N = X, 2x, = 3x and…

A:

Q: Show that 1 0 1 0 0 1 0 1 00 1 det: ⠀⠀⠀ 000 1000 0 0 00 00 ⠀ ⠀| = 1+ (−1)”+1. 01

A:

Q: 16) n = 4; 2, 1 and 3 + 3i are zeros; f(1) =78 %3D

A:

Q: Is Q closed over multiplication? (Y/N)

A:

Q: (- 1)"x" Σ 6" (n² + 9) n=1 8.

A: Given

Q: Consider the number C(n,r) given by n! C(n, r) = (2.1) (n – r) ! -r ! where n and r are integers…

A: Given : Cn, r is given by      Cn, r=n!n-r!·r! where n and r are integers satisfying      n≥2    and…

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: all n N we have We apply this successively for n = bn+1 bn N, N+1, n 9 bN+1 < (m + €)bN bN+2 < (m+…

A:

Q: Q.4) Find the value of Sigma (d=sum of student id + 22, and c=12) 4c(d + x)(7c* +2cz +9z*) ((d +x)*…

A: As the student ID is not provided, I assume the sum of the digits are 10 (say). Then d=10+22=32 and…

Q: Σ s2 an convergent or divergent?

A:

Q: has its largest value. a = Find two numbers a and b with a ≤ b such that b= # # L (6-52 (6-5x-x²)…

A: Given integral is ∫ab6−5x−x213dx. We have to find the value of a and b with a≤b. The integral…

Q: Prove E-1(4i – 3) = n(2n – 1) for natural numbers n 2 1. i=1

A:

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

A:

Q: Compute (a) E-ok · k! = (n+ 1)! – 1. (b) D (") = (). k=0 TIL

A: The problem can be solved using concepts of permutation and combination.

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

A:

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

A:

Q: 1 = (₁ + - +/- ) ² (b) Prove that n < 1+ for all n € N.

A:

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: 34) Ź (K²_2K) is e (n3) Ź ķ? ビ=3 Ź 2K = k=3 ntro)(2n+1), entntt) Sungs are さ。 2n3+ 3n?+n – ant- Gn…

A: We have solve the given  problem

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 ≥…

A:

Q: 4(-1)" Σ 3) Estimate 2N 1 correct to one decimal place. N-1

A: