(1) show that 21 divides 4+1 +52n-1 for n ≥ 1

Q: How do we know that f(n) has an equal growth rate to g(n)

A: for that let's understand what is thita: It is defined as the tightest bound and the tightest bound…

Q: Suppose that f(n) ≤ 5g (n) for all n. Then we can certainly conclude that Of(n) = (g(n)) Of(n) =…

A: Performing a complexity analysis on an algorithm is crucial for developing new ones. To a large…

Q: Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n,…

A: Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n,…

Q: Solve completely for a(n) a(n+2) = 3a(n+1)-2a(n) a(0)-3 a(1) =4

A: This is recurrence relation problem which uses distinct roots theorem.

Q: Use the master theorem to give an asymptotic tight bound for the following recurrences. Tell me the…

A: The master theorem is a technique for analyzing the time complexity of recursive algorithms. It…

Q: 5. Define the following (almost Fibonacci) recurrence for n = 0 Gn for n = 1 Gn-1+Gn-2+1 for n2 2…

A: Hey there, I am writing the required solution of the questin mentioned above. Please do find the…

Q: solve the following recurrence using master theorem T(n)=81T(n/3)+n^4

A: In this question we have given a recurrence and we need to find the solution of the recurrence using…

Q: Solve the recurrence: T(n) = T(n/2) + 4n T(1) = 1

A: The solution to the given problem is below.

Q: Based on the master theorem, what is the solution to T (n) = 3T (2/2) + n² (n²) oe (n² logn) (n³) e…

A: Master's theorem: To apply master's theorem, the recurrence relation must be in this form: T(n) =…

Q: Wilson's Theorem states that for any natural number n >1, n is prime if and only if (n – 1)! = -1…

A: Please check out the code and output screenshots below-

Q: (3) Let (an: n € N} be a sequence of real numbers. Prove that lim sup, lan = 0 if and only if an → 0…

A:

Q: Proof that (11^n) −6 is divisible by 5 for all values of n ≥1

A: We have to prove that (11^n) −6 is divisible by 5 for all values of n ≥1. Using Induction method we…

Q: T(n) = 2T(n/4) + nis O(n).

A: the solution of T(n) = 2T(n/4) + n is O(n).

Q: [Problem 2] Using substitution method only, prove that a) T(n) = 7T(n/8) + 2n is O(n) b) T(n)=…

A: We need to prove the given relations using substitution method.

Q: Prove √15n2log(5n) + 0.4n +√n is Θ(n2log(n)) using definitions only

A:

Q: 15- Prove that for every positive integer n: 1·2· 3 + 2·3 - 4+ . +n(n + 1)(n + 2) = n(n + 1)(n +…

A: principle of mathematical induction.    Given p(n) checking for n=1,    LHS:1.2.3=6…

Q: 1 + 3 + 5 + ....... + (2n-1) = n^2 P(k+1) Prove the induction

A: Induction Proof: It is used to prove the particular sequence is true. There are three steps in this…

Q: c) Prove that the following claim holds when for all n ≥1 Σ(8i-5)=4n²- n i=1 n

A: SOLUTION - In this problem, we need to prove : ∑i=1n(8i−5)=4n2−n   PROOF -   Taking left-hand side,…

Q: Based on the master theorem, what is the solution to T (n) = 16 (7) + n ○e (n³) e (n² log n) ○e (¹)…

A: Answer is:  Θ(n2)     The running time of divide and conquer algorithms may be calculated using the…

Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.