(a) Show that 1++++ 1 21+, 2n-1 2n whenever n is a nonnegative integer.

Q: Prove that (-1ti3)" + (-1-is3)" has either the value 2n+' or the value -2" if r is any ). integer…

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Q: Prove the bi-conditional: prove that if 5n+2 is even, then n is even, where n is an integer.

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Q: Use mathematical induction to prove that the statement is true for every positive integer n: 2 + 6 +…

A: To prove 2+6+10+....4n-2=2n2

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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Q: Let n be a positive integer. Prove that 1:(1)+2 (;) . (:)- n n + 2 . +n . n2"-1. ... n

A: We have to prove the identity: 1·n1+2·n2+....+n·nn=n2n-1

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: 6. Let r> 2 be an integer. Show that R(2r, 2r) > 2

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Q: Prove that the statement is true for every positive integer n.

A: 11·2+12·3+13·4+---+1nn+1=nn+1 Let us prove this statement by using the Principle of Mathematical…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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

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Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: Prove that for any nonnegative integer n. n + +.. + 22m i+1 i +2 n+1 i=0 the

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Let n be a positive integer. Prove that (:) () (:) 1. 1 + 2. 2 +n n2"-1

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Q: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: 10. Prove that the product of two integers is odd if and only if both of the integers are odd. 11.…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove that the statement is true for every positive integer n.

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: Prove that n(n + 1)(2n + 1) (3n2 + 3n - 1) 30 whenever n is a positive integer.

A: see below attached explanation

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

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

Q: 85. (a) Prove that any integer is expressible in the form 4n, 4n + 1, 4n + 2, or 4n + 3 where n is…

A: Here we use mainly dividend = divisor × quotient + remainder.

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: Use mathematical induction to prove that 2n+2 + 32n+1 is divisible by 7 for all integers n 2 0.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

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Q: 1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Prove that 3n^2 + 1 is even if and only if n is an odd integer.

A: We need to prove that 3n2+1 is even if and only if n is an odd integer.

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n. 22+42+62+...+2n2=2n2n+1n+13 We prove…

Q: Using "Proof by Contradiction" to show that, if 2n³ 5n is odd thenn is an even integer.

A: Now, we have to show that if 2n3-5n is odd then n is an even integer by contradiction. Now, Lets…

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: Prove that 1 + 3 + • • • + (2n - 1) = n2 for all natural numbers n.

A: Our Aim is to prove that 1 + 3 + .... + (2n - 1) = n2 for all natural numbers n.

Q: 2""(-1)" +1 2. Prove that 1-2+2² – 2³ +...+(-1)" 2" =- for all positive integers n.

A: We prove this result by using mathematical induction. For answer please look at step 2.

Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: Use mathematical induction to prove the inequalities. Prove that n2 - 7n + 12 is nonnegative…

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Q: Prove that 1- 1! +2 · 2! + .. +n ·n! = (n+ 1)! – 1 whenever n is a positive integer.

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Q: Let n be an odd positive integer. Find all perfect numbers of the form nn +1.

A: n=1,3,5,7,9,11,13,...nn+1=11+1,33+1,55+1,77+1,99+1,... Sequence is: 2, 28, 3126, 823544, 387420490,…

Q: Prove that there is an infinity of natural numbers n such that 2n +1 is divisible by n3

A: Solution of the problem as follows

Q: Prove that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…