The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is (b) n² n(n - 1) n(n +1) (d) (а) n 2

Q: u will prove that 7*+1 - 1 is divisible by 6. ypothesis, since 6 evenly divides , there exists an…

A: We will fill in the blanks.

Q: (b) Use mathematical induction method to show that 품(주-)- (두-0) (주-)-(-)1 n+1 2n

A:

Q: Establish by mathematical induction that any postage of at least 12c can be obtained using 3c and 7c…

A: We have to establish by mathematical induction that any postage of at least 12c can be obtainedusing…

Q: Prove by the principle of mathematical induction that 1 x 1!+2 x2! + 3 x 3! +…+ n x n!= (n+1)!—1…

A: to prove the result 1×1!+2×2!+3×3!+...+n×n!=n+1!-1 for all natural numbers n , proceed as follows…

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: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: Let k, n be positive integers with k < n. Give a combinatorial proof of the identity (in the form…

A:

Q: Use the principle of mathematical induction to prove that (1 x 4) + (2 x 5) + (3 x 6) + .+ n x (n…

A: Given expression is We will prove the result by principle of mathematical induction Step1. For n=1…

Q: Prove that 5n-1 is divisible by 4 to any positive integer n

A:

Q: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

A:

Q: The general combinatorial formula for the nth triangular number (1, 3, 6, 10, 15, 21, .) is On+1Cn…

A:

Q: Determine the smallest subring of Q that contains 2/3.

A:

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Prove by cases that n2−1 is divisible by 3 when n is an integer not divisible by 3.

A: The detailed solution is given below Thank you

Q: Suppose a, n E Z and 0 < a< n. Prove that ()- (":")- (:-) n - 1 %3D + a a а- 1 in two different…

A: Given a,n∈ℤ such that 0&lt;a&lt;n. (1) Consider:…

Q: By Principle of Mathematical Induction prove 6n 6

A:

Q: With the help of principle of mathematical induction, prove that 1 * 2 + 3 * 4 + 5 * 6 +…….+…

A:

Q: How many elements are there in the greatest conjugacy class of 5 ?S

A:

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

A:

Q: Use mathematical induction to show that Σ (4k – 1) = n(2n + 1) %3D k=1

A: To prove the given statement by using mathematical induction.

Q: Use Fermat's Theorem to show that n 13 - n is divisible by 2730 for all n.

A:

Q: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

A:

Q: Show that there are infinitely many primes of the form 6k + 5.

A:

Q: 5 (9 9 () 2. Find a reduced residue system modulo 2", where m is a positive integer. in a reduced…

A: To find a reduced residue system modulo 2m, where m is a positive integer.

Q: Number theory

A: Consider the provided question,

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

A:

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Show that for every integer n, f (n) = (2 n3 + 3 n – 10 n + 5 n) is divisible by 7. %3D (Hint: use…

A:

Q: Prove that n² +n is divisible by 2, where n E N.

A:

Q: 2. Prove by the Principle of Mathematical Induction that 5-2 + 5-4 + 5-6 +... + 5-2n = 24-'(1 –…

A:

Q: Use combinatorial proof to show that 10 (2n - 5 3 n - 2 n | %3D 10 k 10 – k

A: Given:

Q: Compute the supremum and infimum of the set A = {n/(2n+1) : n ∈ ℕ}

A: We have to find the supremum and infimum of the following set A =n(2n+1):n∈ℕ

Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

A:

Q: 5. Invent a positive integer a that has at least 8 decimal digits and a ones digit of 7. Estimate…

A:

Q: Prove that n2+n  is  divisible by 2, where nN.

A:

Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

A:

Q: How many elements of order 4 does Z4 ⊕ Z4 have? (Do not do this by examining each element.) Explain…

A:

Q: Prove that: there exist infinitely many 5-tuples of practical numbers of the form (n-8, n – 4, n,…

A: It is given that 5-tuples of practical number of the form (n-8, n-4, n, n+4, n+8)

Q: Prove by the Principle of Mathematical Induction that 1(1!) + 2(2!) + 3(3!) + .. +n(n!) = (n+1)!- 1…

A:

Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

A:

Q: 6. By Principle of Mathematical Induction prove 6n 6

A:

Q: (1-) (1--)--(1-=) 1 n+1 2.

A:

Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

A:

Q: Obtain the expansion of ; up to the term in x* 1-3x2

A: We know 11-x = 1+x+x2+x3+ ⋯

Q: Provide a combinatorial proof that includes quarternary strings {0,1,2,3} to prove that: (:) - () +…

A:

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A:

Q: By Principle of Mathematical Induction, prove 21+ 22+ ... 2" = 2n+1-2 for all n>1 %3D

A: