Let n be any integer. Prove that 3n? – 2 is odd if and only if n is odd.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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

A: Given, 3+9+15+..+(6n-3)=3n2

Q: If n2 4 is an integer,prove by induction that n<nl.

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

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

Q: Let n be any integer. Prove that 3n - 2 is odd if and only if n is odd. Suppose x and y are any…

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Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contradiction.

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

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: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Prove that for all integers a, b, and c if a divides b then for all naturla numbers n a^n divides…

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Q: ! Prove the following statements. (a) If k is a positive integer, then 3k+2 and k+1 are relatively…

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Q: Prove that the following statement is true for all positive integers m and n:  m and n are…

A: m and n are multiples of each other if and only if m = n. for all positive integers m and n:

Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

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

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

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 8 | (54" +7) for every positive integer n.

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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

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

Q: Use proof by contraposition to prove that the following statement is true for all positive integers…

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Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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

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Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

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Q: For all integers n>0, prove: 2n < (n+2)!

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Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Prove that “if n3 + 5 is odd, then n is even” by using the following: a) Direct Proof b) Indirect…

A: If n3 +5 is odd then n is even. Given:  n3 +5 = 2K+1                (K is any integer)       n3  =…

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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Q: Prove by induction that for every integer n 2 1, (xe2*) = 2"-1(2x + n)e", ne Z

A: We prove this by Mathematical induction

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: Prove that a positive integer n is p

A: Topic :- Number system

Q: Q. Prove that by mathematically Induction 2" < (n + 1)! , for all integers n 2 2. а.

A: We Solve it by First Principle of Mathematical Induction.

Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

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Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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Q: Prove that the statements here are true for every positive integer n. 12 + 22 + 32 + · · · + n2=n(n…

A: Proved that for n≥1 fn=12+22+32+⋯⋯+n2=nn+12n+16 For n=1 LHS=12=1 RHS=nn+12n+16=11+12+16=66=1…

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

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…

Q: Prove that for any integer n, if n³ is odd then n is odd.

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Q: Prove by contradiction that 6n + 5 is odd for all integers n.

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

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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Q: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

A: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

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

A: a&gt;1, then an&gt;1

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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