For every real number x+1, n-1 1+x+x² +x' +. x´+x' +...+x" for all neN. 1-x

Q: 1. Prove that 7(x – 1) 2.

A: Let, f(x)= ex-7x+7 ,for x≥2. Differentiating both sides with respect to x, f'(x)= ex-7≥0 . This…

Q: Use mathematical Inductlon to prove that for r>0 and any positive Integer

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Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

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Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

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Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

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Q: Prove Bernoulli's Identity: For every real numberx > -1 and every positive integer n, (1+x)" > 1+nx.

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Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

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Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

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Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: Show that yn for all n=1, 2₁- ✓

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Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

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

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Prove n (2n = n n 1 k=1

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Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: (":')-(:)·(,".) n +1 n n - Prove that for all integers n > i > 1. i 1

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Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

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Q: 7. Prove that 2. O = (-1). %3D

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Q: Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: Prove That Cn) =nsh ツールがー

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Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

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Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: 5) Prove that d tanh ) |-x2

A: Prove that  d/dx[tanh-1x] =1/(1-x2)

Q: part ii and iii, thank you

A: According to pascal law,

Q: The sum of any two integers of the from 4k + 1 is always: Even O prime Zero Odd

A: If a number n is not divisible by 2 then it is an odd number.  If a number n divisible by 2 then it…

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

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Q: coth- x = In () Q1/ Prove that

A: To prove coth-1x=(1/2)ln[(x+1)/(x-1)]

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

A: We prove that a positive integer n is even if and only if n2 is even.

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: 1) Prove that 0|a - a=0

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Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: For n2 1. prove using mathematical induction: 2" s 2n+1 - 2n-1 - 1

A: We have to solve given problem:

Q: Prove that le-2z| 0.

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Q: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.