62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer, then Ek = Use a method similar to the one for the previous exercise, but take note of what happens with the extra middle term of the sum.) (Hint: 2

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A: The solution is given as

Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: Prove that the hundreds digit of n20 is even.

A: To Prove that the hundreds digit of n20 is even.

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A:

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A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

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A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² + n 2 Sn =…

A: Given:  Sn= 1+2+3+.......................+n = n2+n2and,Qn=1.1+2.2+3.3+...............+n.n=n33+n22+n6

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

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A:

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Q: 1. Show that if m and n are odd integers, then m+n is an even intege

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Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

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A: Sum of n number is given by

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² Sn…

A: This question is related to principal of Mathematical induction . solve in step 2.

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A:

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A:

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A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

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A:

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Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

A:

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A:

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Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

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Q: 2. Prove that a positive integer n is even if and only if nis even.

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A:

Q: 1. а) Suppose that fis defined recursively by: f(0) = 5 and f(n+1) = 2 fin + 5. Find (1). A2). A3)…

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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: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…