Use strong induction to show that every positive integer can be written as a sum of distinct powers of two, that is, as a sum of a subset of the integers 20 =1, 21 =2, 22 =4, and so on. [Hint: For the inductive step, separately consider the case where k + 1 is even and where it is odd. When it is even, note that (k + 1)∕2 is an integer.]

Q: Every integer is even or odd. Use induction.

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Q: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

Q: Jse a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: Decide which of the following are valid proofs of the following statement: Let n be any integer.…

A: Let n be any integer. Then n3-n is divisible by 3.

Q: Use induction to prove the following for all natural numbers n: 1²+2²+3²+ n(n+1) (2n+1) +n n² = 6…

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Q: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: n2 - 2 is

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Q: Use induction to show that 1+3+5+...+(2n-1) - n for all natural numbers n.

A: We will use the method of induction to show that 1 + 3 + 5 + ... + (2n-1) = n2 for all natural…

Q: Use the method of mathematical induction to prove the following statement where n is a positive…

A: Mathematical induction.....