Suppose we prove that 2" > n? for n=1 and 2. Knowing that the inequality is true for some n n > 4 and we can show that it must be true for n+1 instead of n, can we conclude that 2" > n2 is true for all n > 1 ? True False

Q: (b) For all integers n, if n² is odd, then n is also odd.

A: Given statement: For all integers, if n2 is odd, then n is also odd. Prove this statement by…

Q: - Use mathematical induction to prove the following inequality (1+x) ≥ 1+ Nx when x>-1 for all…

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Q: Question 1:  How many ways can they line up so there is always atleast 1 duck between any two geese.

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Q: The following "Theorem" is obviously not true. Explain what's wrong with the proof. Theorem: 1 is…

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Q: How many integer solutions to ₁ + x₂ + x3 + x₁ = 28 are there for which ₁ ≥ 6, ₂ ≥ 6, 3 ≥ 3 and 4 ≥…

A: y1=x1−6y2=x2−6y3=x3−3y4=x4−1…

Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: 4. Knowing that 1 + 3 + 5 + …+ (2n – 1) = n², find 1++3 +5 + .…+ 135 =

A: Solution :   Given that  1 + 3 + 5 + ... + (2n - 1) = n2    -----(1)

Q: N is less than 60. All the factors of N are bigger than 60. How many natural numbers can be N?

A: It is given that N is less than 60. N<60

Q: Suppose V₁,..., Vm € V. Prove that {v₁, ..., vm}¹ = (span(v₁, ..., Vm))¹.

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Q: For all integers n, if 6 | n, then 9 | ((n^2) + 21n/2)

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Q: This is a mathematics question. There are 8 numbers in a sequence that add up to 250 (ex.…

A: The details are given as step 2.

Q: For how many integers n with 0 ≤ n ≤ 100 is the quantity integer? n(n-1) (2n-1) 12 equal to an

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Q: How would you solve this using direct proof

A: Step 1: Recall some definitions.  Step 2: Given informations. Step 3: Proof of the statement.

Q: • Show that if m divides 3, then m divides 9. (We say a divides b if there exists an integer k such…

A: Let  m divides 3 .  Then there exists an integer k such that 3 = mk. Now, 9 can be written as…

Q: How do I prove: If n is a natural number and d|n prove that d ≤ n.

A: How do I prove: If n is a natural number and d|n prove that d ≤ n.

Q: Find the integer in the interval (7,10) with the property that if we divide it by 4, the result is…

A: Solve the following

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A: The objective is to test the claim that the majority of precalculus students are passing the class…

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Q: k is a prime number that has a ones digit of 7 and 0 <k < 100, how many values of k are possible?

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Q: Please help me to write this as a proof.  Let n be an integer. If n is the sum of three consecutive…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.