If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6 2) 8 3) 10

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

A:

Q: *4. Prove that 1³ +2³+ ... + n² = n²(n+1)² for all ne N.

A: Introduction: A strong and elegant method for proving specific kinds of mathematical claims is…

Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

A:

Q: Show that for all positive integers n, 13 + 2° + ...+ n³ = (1+2+3+ ...+ n)?

A:

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: what

A:

Q: 5. Justify by mathematical induction that for every positive integer n, 1(1!) + 2(2!) + 3(3!) + · .…

A:

Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

A:

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

A:

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

A:

Q: Argue by contradiction that for all integers n > 2, there is a prime number p such that n<p<n!,…

A: We have to prove that there is a prime number p such that n &lt; p &lt; n!, where n &gt; 2 and n! =…

Q: ) Prove every integer n >= 7, n! > 3n

A:

Q: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

A:

Q: Prove that 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = (3^(n+1) -3)/(2) for every n ∈ N

A:

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: If n is an integer and 3n + 2 is even then n is even. (You may use any kind of proof)

A: Given: To prove that if n is an integer and 3n+2 is even then n is even.

Q: 2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

A: Concept:

Q: 7. By Principle of Mathematical Induction prove 5º – 1 is divisible by 2 for all n>1

A:

Q: P(n): 2"+1 > 2n + 1

A:

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

A:

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Suppose there is a currency which only uses 3-cent coins and 7-cent coins. Prove that for all…

A:

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

A:

Q: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A:

Q: 5. Justify by mathematical induction that for every positive integer n, 1(1!) +2(2!) + 3(3!) + .…

A: Given: 1(1!)+2(2!)+3(3!)+.....n(n!)=(n+1)!-1

Q: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: 2. Find a counterexample to the statement n2 – 4 - V nonzero integers n, is not an integer.

A: 2. Given statement is, ∀ nonzero integers n, n2-4n is not an integer

Q: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: Prove that n^2+5 >=4n for all integers, n>0.

A:

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: 5. Prove by the principte of mathematical induction. show that 1 +2+2² +.+ 2" - 2"1/for all non…

A: To prove1+2+22+2+...+2n=2n+1-1 using mathematical induction.

Q: 74. P is a natural number of at least 6 digits and its left- most digit is 7. When this leftmost…

A: Introductions :

Q: 2. Use mathematical induction to prove 8^n - 1 is a multiple of 7 whenever n is a positive integer.

A: The given problem is to prove the following statement using the mathematical induction method. Given…