1. Prove that for all numbers of k, (5*2^(2k+1) - 1)/3 is an odd number.

Q: find all integers n>1 such that 1^n+2^n+.....+(n-1)^n is divisible by n.

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Q: Prove that 7^n − 1 is a multiple of 6 for all n ∈ N. ( N is Natural Number)

A: Advance maths

Q: n be a positive natural nun (an)" is equivalent to: 4n Let a) n²n (n²-4n) b) n(n²^_^) d) na^ (^^-2^)…

A: Given, n4n-(2n)2n

Q: Rationalize the denomi 20 5/10 - 7/5

A: 20510-75

Q: integer greater than 1 can be written in the form n = p₁ p2p with a_{i} > 0 and p, are distinct…

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: Prove that for all integers, n , n^3-n is divisible by 3.

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Q: For 11- 15, Given f(x ) = 2x + 4, g(x) = x^2 -4 and h(x) = x^2+2x+111. What is f(3) = _? * Your…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Prove that 1 + nx is less than or equal to (1+x)^n for all real numbers x>-1 and integers n >=2…

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Q: Write a formal induction proof (Base Case and Inductive Step) Prove by induction that…

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Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

A: SOLUTION-

Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

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Q: Prove or disprove the following statement: There are prime numbers a, b, c such that a^2+ b^2= c^2.

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Q: n>=0

A: According to the question,    False. This statement is not true for all integers n >= 0. A…

Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: K = (perfect squares of integers} under addition? Is it closed or not

A: K is the perfect square of integerNow we find whether it is closed or not.

Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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Q: Prove that for all divides natural numbers nt 7^-2^.

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

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Q: For 7-10 Given a_1= 3, a_2= 7 and a_n= [2a] _(n-1) + a_(n-2) for n 23, 7) Find a_3 =,* Your answer…

A: First Question: (7) Given a1 = 3 , a2 = 7 and  an = 2an-1+an-2 then a3 = 2a2+a1 = 2*7+3 = 17 Hence…

Q: Find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime. Justify that your…

A: We have to find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime.

Q: Prove by induction that for positive integers n, 13|(4^(2n+1)+3^(n-1) )

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Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: Prove that for all natural numbers k, a (4^k - 1)/3 + 2*a*4^k is of the form 1 + 6a

A: We show that the question doesnt hold for a=1 and k=1

Q: = A positive integer is called perfect if it is equal to the sum of its positive divisors other than…

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Q: The 4th term in the expansion of (2x² – 3x)* is - 10 Ca (2r²)*(-3z)® O 10 C4 (2a²)°(–3x)“ O 10 C3…

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Q: Find all positive integers a, b such that a | b^2-b+1 and b | a^2-a+1

A: Solution :-

Q: For n is greater than or equal to  1, use mathematical induction to establish each of the following…

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Q: Show that the weighted arithmetic mean of first 'n' natural numbers whose weight are equal to the…

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Q: While proving (a*2(2a+1 is an integer for the case a = 6q +1 which of the following will be obtained…

A: We are given with a(a+1)(2a+1)6 We need to substitute a=6q+1

Q: Given an odd integer a, establish that a^(2) + (a+2)^(2) + (a+4)^(2) +1 is divisiible by 12 using…

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Q: x+(x+1)+(x+2)+(x+3)+(x+4)=5x+10 What does this prove about the sum of these natural numbers?

A: The sum of the given equation is calculated as follows.

Q: Prove that for all natural numbers k, (4^k - 1)/3 is an odd number.

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