n2 1, n - nis divisible by 3.

Q: Show that there are infinitely many primes.

A:

Q: Use mathematical induction to show that 5? − 1 is divisible by 4. Determine the starting value as…

A:

Q: An theorem that specifies the complete expansion of a binomial raised to any positive integer power…

A: Complete expansion can be done by theorem

Q: What is the sum in sigma notation of this: 8x + 8x2 + 8x3 + ..... + 8xn Thank you.

A: Given that  geometric series ,S = 8x+8x2+8x3+.......8xnS= 8(x+x2+x3+.....xn)

Q: Suppose you draw n lines in the plane so that every pair of lines cross (no lines are parallel) and…

A: Let n be the number of non parallel lines. Given that number of regions= Rn Let Un represent maximum…

Q: Use mathematical induction to prove that 3n²+1 is divisible by 4 whenever n is an odd number.

A:

Q: There are 8 people in a room, whose ages are between 10 and 40. Show that we can always find two…

A: Pigeonhole principle Here, we don't know the ages of 8 people, but we know the range of sum of their…

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: What is the remainder when 2021202120212021 is divided by 2020? use congruence modulo

A: a = 2021202120212021 b = 2020 If a = qb + r then  a ≡ r mod n.

Q: In the expansion of + 6)-", state the number of terms and the coefficient or the middle term. 21, 20…

A: Introduction: The binomial expansion is a very important finite series in mathematics. With the aid…

Q: Pigeonhole Principle Prove that among any 100 integers there always two whose difference is…

A: The given problem is to prove the following statement which follows the pigeonhole principle, we…

Q: Give a formula for the number of permutations of [n] with no cycles of length exactly 2. Your…

A: Formula for the number of permutations of [n] with no cycles of length exactly 2 is derived.

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Given that it is to be found all three-digit numbers with different digits with the property that…

Q: Verify using fast exponentiation that the congruence 12964026 = 15179 (mod 64027) is true. Can you…

A: Check the congruence relation       12964026≡15179 mod 64027 using fast exponentiation. Also…

Q: A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each…

A: Given:     The sequence is:     1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,... To get this number, we have to…

Q: V nEN, 7"- 3n is a multiple of 4.

A: We have to find   For all n= natural number we have to

Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: Use sigma notation to write the sum.

A:

Q: I begin proofs by mathematical induction by writing Sk and Sk+1, both of which I assume to be…

A: you have taken that both kth and (k+1)th as true before.

Q: Discrete Math

A: According to the given information, Suppose that:

Q: By using the principle of mathematical induction, prove that 1² + 2² + 3² + ..... + n² = fraction…

A: In this question, we prove the given statement  12+22+.....   + n2=n(n+1)(2n+1)6 By using…

Q: Let n be a number with 3 digits. (i.e 100sns 999) Show that if the sum of digits of n is divisible…

A:

Q: All composite number can represent as product of primes numbers

A:

Q: Determine the formula and (largest) domain for the following expression and then prove it using…

A:

Q: A sequence generated by the length 3 recurrence starts 001110. Find the next four elements of the…

A: Given that the sequence is generates by the length 3 recurrence relation. Therefore, the recurrence…

Q: Use mathematical induction to prove that any amount of postage that is an integer amount greater…

A: We have to prove that any amount of postage that is an integer amount greater than 53 cents can be…

Q: a) Proe by induction on n3, that 23"-/ is diwisible bu

A:

Q: a sequence generated by the length three recurrence starts 001110 find the next four elements of the…

A:

Q: or an integer n>=3, an n-gon is an n-sided polygon. (example, a 3-gon is a triangle, 4-gon is a…

A: To prove: the sum of the interior anles of every polygon with n sides is n-2180° Now we shall take a…

Q: A knight on a chessboard can move one space horizontally (in either direction) and two spaces…

A: Consider the provided question, We have to proof: The knight can visit every square. Proof by…

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

A:

Q: Show that n-lis divisible by 8, if n is an odd positive integer

A:

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. In a proof by…

A: We have for each positive number n P(n)=5n-1 To prove P(n) is divisible by 4 using mathematical…

Q: Use mathematical induction to prove: For each natural number n, 6 divides (n3-n). I need to write a…

A:

Q: Use mathematical induction to show that given a set of n + 1 positive integers, none exceeding 2n,…

A:

Q: 7. What is the second step In proving by mathematical induction that for every positive Integer n,…

A:

Q: Find the recursive and explicit rule for -1, 3, 7, 11,... Recursive rule

A: This is a problem of series . Based on concepts and formulas we will answer the question.

Q: In two or more complete sentences, explain how to find the third term of the expansion of (2x+y)4.

A:

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

A:

Q: Find the sum of the first five natural even numbers using sigma notation.

A: We have to find sun of first 5 even natural numbers.

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: divisible by 3

A:

Q: (5n)! Prove that 40"-n! is natural number for every natural number n.

A:

Q: Let n be a positive integer. Give a recursive algorithm for computing the sum of the first n…

A: To find - Let n be a positive integer. Give a recursive algorithm for computing the sum of the first…

Q: Using the fast exponentiation method (page 182), determine 535 mod 47. Show your work.

A: We will find out the required value.

Q: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

A: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

Q: A DIGITAL CHIP MADE BY YOUR. MICROELECTRONICS DIVISION, COUNTS CONTINUOUSLY IN THE SEQUENCE: 1024…

A: The counts of the chip are 1024,2048,4096,8192,.... Now, we can see that 20481024=40962048=2 Each…

Q: Use mathematical induction to prove that for every integer n >=2, if a set S has n elements, then…

A: We have to prove that The given statement is for every integer n ≥ 2, if a set S has n elements.

Q: If n> 5 is an integer, prove the identity () - OO(;)-O0C:")- O0(:)--(":)OO +..+ by counting in two…

A: We are given a formal containing sum of binomial coefficients and we have to prove it using two…