1=1,f2 =1 and fn+2 =fn +fn+1 for all n≥1.    Prove that for all natural numbers n≥1: f1 f2 +f2 f3 +f3 f4 +...+f2n-1 f2n =f

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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Q: In(x²) = 2 ln(x) for all x True O False

A: ln(x^2) =2ln(x) for all x

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

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Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

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Q: Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1)…

A: Given, r is a real number and k is an integer  The Binomial Coefficient rk by rk= r (r - 1) --- (r -…

Q: (c) Prove that r| +11 36 T-1 2 +36 for all real I.

A: We have to prove that the following inequality holds: x-11x2+36≤x+1136 for all real x. We will use…

Q: 3. Invent two interesting and original positive integers a and b. Let f(x) = ax +3. (a) (b) What is…

A: Given function is fx=ax+3. To find:  a  f10   (b). A number x such that fx=b Solution: Given…

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

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Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: how that 1+2+2²+23+...+2 = 2n+1 -1 for n≥ 0.

A: We need to prove by induction hypothesis      1+2+22+.........+2n=2n+1-1, for n≥0

Q: Show that when |z +1| < 1,z² =1+£(n +1)(z+1)" n=1

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Q: Using Proof of Contradiction, prove that for all integers x and y, if x + y is odd, then x ≠ y.

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Q: Prove that 5n²+3n+2 € O(n²).

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Q: Let a and b be non-negative real numbers with b+ 0 and p,q > 1 and+= 1, then a show that ab <+

A: Define a function y : [0,∞)→ℝ by yt=1q+tp-t1p, for every t∈ [0,∞). Now differentiating,…

Q: Prove 2+4+8...+2n=2n+1-2

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

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Q: If a, b, c are non-zero real numbers, then bc ca ab ca ab bc vanishes, when ab bc ca ca 1.1 -+- 0…

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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 all integers n>0, prove: 2n < (n+2)!

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

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Q: show that the general factorial function: () (¹) (+).. m (a)m here m is a positive integer and n is…

A: Given Data: To show that the general factorial function  αmn=mmnαmnα+1mn.........α+m-1mn here m is a…

Q: umbers and q (> 0) the first of n harmonic means between the me numbers. Then prove that

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Q: Suppose X₁ ~ N(6, 1) and X₂ ~ N(7,2), and X₁ is independent of X₂. Find P(X₁ < X₂) and P(X₁ < X₂ +…

A: We know linear combination of normal distribution is also a normal distribution. Therefore any…

Q: Find the general solution of the recursive relation an = 2 an-1 - 6 an-2 for every integer n > 1. a,…

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Q: Prove by, Proof by Contradiction.   (a) If n is an integer, then n2+ n3 is an even number.   (b) If…

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Q: Let P(x) denote the statement "x < 4." P(4) is A) True False

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Q: Lef 5={x|(x-)(x-3)< of; T= }x| 12x+1)s a) Show s & T on Ore musndn line. 6) Are S& T or cloed on…

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Q: (b) Prove that 32n²+1, for n ≥ 1.

A: The given inequality is 3n≥2n2+1, for n≥1. We have to prove this inequality.

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.