7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... + an = 0. Prove that ifr is rational, then it is an integer. 8. Prove that for every n > 1, the sum 1 1 1 + + ... 2 is not an integer.

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Q: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

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A: We have to proof by induction process

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A: Consider the provided expression, 5 Show that the above expression is irrational. We proof by…

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A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

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A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: 1. Prove that V3 is irrational number.

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: prove that √2+√3 is irrational

A: For rational we needed form p by q P and q are integer

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A: The product of any two even integers is a multiple of 4.

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Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

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Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

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Q: Prove that 3 divides n3+2n whenever n is a positive integer

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A: Here we will show that given statement is true by mathematical induction method.

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

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A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

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Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Prove that 2+/3 is irrational

A: Given,          To Prove : 2+3 is irrational.

Q: Prove, that (nezond n73) If x"=5,+hen x is not a rational number.

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Q: 7. Suppose a1, a2,...,an are integers, r is a solution to the equation x" + a1x"- + a2x"- +.. + an =…

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: Show that the sum of the first n positive odd integers, 1 + 3 + 5 + .......+ (2n - 1), is n2.

A: Given: 1+3+5+...........+2n-1 is n2

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Q: let m, n exist in the integers. n does not equal 0. prove that if n2 x2 - 2mnx+m2 = n2, then x is…

A: x is a rational number proved directly.

Q: 9. Prove that V3 and /2 are irrational by assuming that each is rational and arguing until a…

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