7. Prove: For neN and F, the nth Fibonacci number, =Fn+1 (1) + (7¹) + ("z ²) + (~³²) +--+ (0) = ² 2 n

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: 22] calculate the number of combination and permutation of, 5 distinct objects taken 2 at time?

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Q: Let p, q, and r be real numbers such that p + q +r = 26 and '/p + lq + '/, = 31. What is the value…

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Q: |: Use Mashematizel Induethon to solve bhe followbing; j I(2") + 2(2²) + 3(2-³)+.. .+ n(2^)…

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Q: : Express as a fraction with no indices.

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Q: True or false: If n is a positive integer with d(n) = 6, then the only possible prime power…

A: n is a positive integer with dn=6.

Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

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Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

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Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

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Q: term

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Q: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

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Q: [a, b] = N=1 (a -,b+-) To prove this, show the following: То Show that [a, b](a – ÷,b +- ) for every…

A: Given that, [a,b]=∩n=1∞a-1n,b+1n This can be written as, ∩n=1∞a-1n,b+1n=[a-1,b] The aim is to show…

Q: For an integer n > 1, how many decimal stings of length n begin or end with an even digit? 5(10)"-1…

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

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Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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Q: 1) Write all the following as expansions : > 0.2 > V26 V35 (0.99)-4 V(4 – 2y)2 ,y<2 2х-1° 2

A: (a) 0.2answer(a)0.2=2100.2=15(b) (26)13 answer(b) using approximation to solve the cube rootput x=27…

Q: 7.) 10² - 15i*

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Q: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: 7. I) Find, if possible, AB ond BA A= 3212/ 2.

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Q: If a and b are integers with gcd(a, b) = d Then gcd d' = -1 d/

A: Given a statement

Q: 13. Express 5 ‹(²) ³ 3 (*√81) ³ as a single power with a base of 9.

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Q: show that 1+1/(3)2+1/(5)2+1/(7)2+...=(pie)2/8 using Eulers pie2/6 equation (show all steps please)

A: The given question is solved below.

Q: 6. For positive integer n, prove the following (given that I(½) = √√): √π = (2n)! √T. 4nn! T(n +…

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Q: п(п+1)(п+2) E}=1jG+ 1) = а. b. Prove that the sum of the first n Fibonacci numbers with odd indices…

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Q: In 3 - In x - In(x + 5) = 0

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Q: Find a formula for +(−1)n. (2). 1 1 ¹ – ²2 ( ² ) + ² ( 2 ) - -· ··- 1 n+1

A: To Find: A formula for .

Q: To show B(m,n)=+x²² -dx (1+x)"¹"

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Q: Prove that 3! 2(n+3) = = n r=0 (-1) 1 n "Cr r+3 Cr

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Q: . Let n be a positive integer. Prove that 1 · ( n/ 1)  + 2 · ( n/ 2)  + . . . + n ·  (n/ n) …

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Q: 10.2 Find a number a that has the following three properties: (i) a = 73003 (mod 3750). (ii) 1 ≤ a ≤…

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Q: One of the striking properties of Fibonacci numbers is that gcd(Fm, Fn) = Fd, (*) where d = gcd (m,…

A: 1. We see,        F9 = 34    [Fibonacci series : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,. . . . . ]…

Q: In x2 10) + In (9) = In (10)

A: Definition used-             ln(x)+ln(y)=ln(xy)

Q: Prove that for each odd natural number n with n greater than or equal to three,…

A: We have a product line called [(1 + frac12)]. (1 - \frac{1}{3}) (1 + \frac{1}{4}) [ ldots left(1 +…

Q: Prove that 3! 2(n+3) = n r=0 (-1) n "Cr r+3 Cr

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Q: n —(2) - (-) (1)¹(1)(). (n-1)

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Q: Show that Consider the Fibonacci numbers {F} defined by F₁ = 1, F₂ = 1, and Fn+2= Fn+1 + Fn. - Fn =…

A: Introduction: In mathematics, a recurrence relation is a significant connection that expresses the…

Q: 5. Prove that the following are irrational: E-2-3', £-12¬!.

A: Expand the given functions and then determine their values. If the values are irrational then the…