A sequence a0, a1, a2, a3, ..., an, .... is define by 7an-1 - 10an-2, for n 2 2 and ao=5 and a,=16. an %3D Prove that an = 3 x 2" +2 x 5" for all integers n 2 0.

Q: an 2. (14 points) Use mathematical induction to prove that a,, = -5+n(n+1), -8 for all integers n >…

A: To prove that 𝑎n = −5+𝑛(𝑛+1) for all integers 𝑛>1 is an explicit formula for the sequence…

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Find the smallest positive integer n such that 5 (3² +2²) (3ª +2¹) (3³ +2³)... (3² +22") > 9256

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Q: Show that for positive integer n,

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Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: 2(5) – 3(1 – 6) 1. Is even integer? 5 2. List all even integers between -6 and 6. 3. Is 124/6…

A: Four questions are given. The solution to the four questions are given below.

Q: Consider the following statement. 9n<5^n   What is the smallest positive integer n for which the…

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Q: 1+ 3n4" for every integer n ≥ 0.

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Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

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Q: Please help me with apply the inductive hypothesis to fk and fk-1 and complete the proof.

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: The number sequence a1, a2, a3, .... is defined by a1 = 8 a2 = 24 and recursion formula 4an + 1…

A: We need to show that an=5n-12n-2 for a1=8 and a2=24  The given recursion formula is…

Q: Prove the following using either direct proof, indirect proof, or proof by contradiction. 1. If n…

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Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

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

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: 3 4 N + 1 5) Show that the sequence 7 2N + 1 is bounded and monotonic.

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Q: If the sequence a1, 92, 93, an 2 az-az +az-a +...+ a²n-1-a²n ... ... is an A.P., then prove that n…

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Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

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Q: A sequence bo, b1, b2, ... is defined by letting bo = 5 and b = 4+bk-1 for each integer k> 1. Show…

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Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: For every integer n > 0, 1+2+22 + · + 2" = 2"+1 – 1.

A: Given: The statement 1+2+22+⋯+2n=2n+1-1 Proof: Base step: Show the statement is true when n=0.…

Q: Prove that 1 + 3 + 5 +···+ (2n − 1) = n2 for every positive integer n (a) by adding 1 + 3 + 5 +···+…

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Q: Prove this: -n is a integer; If 6| n then 18| n^2 +3n

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Q: - Find the summation of a2 + 2a2 + 3a2 + …+ n * a2. ...

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Q: of all positive integers Find the sum less than 200

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Q: 5. Prove that n² is an even number if and only if 3n + 1 is an odd number by proving a) If n² is…

A: To Prove: We have to prove that n2 is an even number if and only if 3n+1 is an odd number.

Q: Show that the sequence delened by +5 mcreasing and ancg"for allkn an is thereas and anz9

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Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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Q: 2(5) – 3(1 – 6) Is even integer?

A: If an integer is divisible by 2, then it is called an even integer. For example, 2,4,6 etc. are even…

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!&gt;3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: Suppose that b1, b2, b3, . . is a sequence defined as = 12, b = br–1 + b-2 for each integer k > 3.…

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