Prove there are no natural numbers n and p with n 2 2 and p² + 1 = 2".

Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

A:

Q: (26) Are there natural numbers n, m such that 20" = 50m? Justify your answer with a proof or a…

A: The idea is to use a prime factorization of the numbers 20 and 50 and then use this to solve the…

Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

A:

Q: nd the smallest positive integer N such that n+2 3n-4 -}| < < 0.01 for all n ≥ N.

A: We want to find N such that n+23n-1 - 13 &lt; 0.01    for all   n ≥ N

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Prove there are no natural numbers n and p with n ≥ 2 and p² + 1 = 2”.

A:

Q: The sum of the first n natural numbers, S = 1 + 2 + 3 + first n natural numbers is 171, determine…

A: Consider the given information:

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

A:

Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

A:

Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: S²₁ Sinmxsinhxbx = 2 x if m=n So min

A: To prove:- ∫-ππsinmxsinnxdx=0if m≠nπif m=n

Q: 4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then 2" – 1 = 127 is a…

A: Solution

Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

A: The objective is to select the correct option from the given options. Given that, for any real…

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: Find the smallest positive integer n such that 5(3² +2²) (3¹ +2¹) (3³ +2°)... (32" +22")> 924 9256

A: Using mathematical induction we can find the solution. Smallest n=9.