Let n be a positive integer such that its prime power factorization n = k2 ... p,", then Ean µ(d)o(d) = %3D

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

A: Divergence of the series

Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

Q: Prove that if n is a multiple of 4, then Fn is a multiple of 3.  

A:

Q: Use mathematical induction to prove that the conjecture P(n) below is true for n e Z* n(n + 1) E (21…

A:

Q: 2) Prove by induction on n that, for all positive integers n: Σ 6(5n6" – 6" + 1) i6 25

A:

Q: Let yk = k", where n is a positive integer; then Ak" = (k + 1)" – k" - (1.112) nk"-+ n kn-2 +...+ 2.…

A: We have to solve given problem:

Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

A:

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

A:

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

A: Formula: nr=n!r!n-r!

Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

A:

Q: Show that n!> n3 if n is large enough.

A:

Q: 1 Prove that + n is an integer for any n E N. 6

A:

Q: Use mathematical induction to prove that: 3n ≥ 2n + 1 for n ≥ 1 discrete math

A:

Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

A:

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: b) Prove that 4" 10 by induction

A:

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: Find a Big-O estimate of (2n+n2)(n3+3n)

A: Given the given expression is 2n+n2n3+3n. Big-O notation: the fn is Ogn for positive constant C and…

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: (i) Give a definition of Fermat's Little Theorem. (ii) For all n e Z, show that 30| (n³ – n).

A:

Q: Prove the following formula for all n E N by induction: 1+ 2·2 + ... +n· 2"-l = 1 + (n – 1)2".

A:

Q: Let yk = k", where n is a positive integer; then Ak" = (k + 1)" – k" %3D (1.112) n = nk"-1 kn~2…

A:

Q: Use Mathematical Induction to prove that for n > 1, n²(n+1)? No other method of proof is acceptable.…

A: We have to prove ∑i=1ni3=n2n+124 by using principal of mathematical induction. First check for  n=1…

Q: If n > 2 is prime then n | a"-1 1 for all positive integers a with gcd(a,n) = 1. Select one: O True…

A: True

Q: Use induction to prove that n п(п+1) Σ- k for all n e N. 2 k=0

A:

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

A:

Q: use mathematical induction to prove, for n > 1: 4n3 E(2k – 1)? = | 3 k=1

A:

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

A:

Q: Prove by mathematical induction that the sum of the first n even positive integers is n' +(2n) = n^2…

A: to prove 2+4+6+..2n=n2+n by induction  Prove the result is true for n=1 LHS  LHS = 2 RHS…

Q: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

A:

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

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

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: For all n E N,n > 1, there exists a prime p such that p|n. صواب İhi O

A:

Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

A:

Q: Let the characteristic of F be n =/0 . Then n is prime.

A:

Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

A:

Q: 10. Prove by mathematical induction that E k (k + 1) = n (n + 1)(n + 2)/3 1sksn 11. Prove by…

A: Given expression: (10)                                                   First , we check if it is…

Q: Use mathematical induction to prove that the given statement is true for all positive integers n.…

A: n3+5n  is a multiple of 6for n=1P(1): (1)3+5(1) is a multiple of 6           1+5   is a multiple of…

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

A:

Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

A:

Q: | +スt3+ *n = nintl) for all 1 てU

A: The given formula is                                    1+2+3+..........+n=n(n+1)2       for all n≥1…

Q: Let a, be the n th digit after the decimal point in 2n + 6e. Evaluate an(.1)". n=1

A: in given problem we have to evaluate :

Q: Use mathematical induction to prove that for z>0 and any positive integer n, e21+z+ 2! n!

A: Recall that,    If f'(x)≤0 in [a,b] then f(b)≤f(a)  and   If f'(x)≥0 in [a,b] then  f(b)≥f(a)

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: Use mathematical induction to prove that:   3n ≥ 2n + 1 for n ≥ 1

A:

Q: 3. Prove that if p is a prime >3 then the numerator of 1 + 1/2+1/3+...+V/(p-1) is divisible by…

A: Given: Let us consider the given, r=1+12+13+...+1p-1

Q: 2) use the mathematical induction to prove : A) 4" - 1 divided by 3 , Vn E N+

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→∞).

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.