Use mathematical induction to prove the statement.
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Answer the given question with a proper explanation and step-by-step solution.
![Use mathematical induction to prove the statement.
23 [n(n+¹)]², for all integers n ≥ 1.
1³ + 2³ +3³ + ... + n³ =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01d01b29-ede7-42ba-8dcd-8c0b18e94cc8%2F02ed25eb-f4d8-40cd-93dc-d8ab06554e43%2Fchk93a4_processed.png&w=3840&q=75)
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- By using recursive approach perform the following first 3 problems by using functions in Java Anlyse their working mechanisms and check their correctness Comment on their T(n)Prove the following statement by contradiction.If n is an integer and n3 + 5 is odd, then n is even.Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod n
- 6. Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1.7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).Solve the following recurrences using iteration methods and Master's Theorem (if possible) a. T(n) = 2T (n/3) +3 b. T(n) = 3T (n/6) + n
- Let P(n) be the statement: “10n – 1 is divisible by 11, where n is a non-zero even natural number” Use mathematical induction to prove P(n). You must show all the steps of the proof using mathematical induction. Show your work step by step.Use two ways to count the number of r-combinations of[n]={1,2,…,n}that contains 1 or 2 or 3. First, separate cases with Case 1 counting all r-combinations of [n] that contains 1 , Case 2 counting all r-combination of [n] that contains 2 but not 1 , and Case 3 counting allrcombination of [n] that contains 3 but not 1 or 2 . Second, count all r-combinations of [n] not containing any of1,2,3and use the subtraction rule.Prove the following statement for positive integers n is odd if and only if 3n2+4 is odd.