16- Prove that for every positive integer n and all real number r except 1: -1 r -1 i=0 17- Show that n! > 3 n for n> 7 (using mathematical Induction )?

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Prove by mathematical induction that for every positive integer n, 13 + 23 + 33 + .+ n³ n²(n+1)² is…

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Q: Use mathematical induction to prove that for all natural numbers n, 3^n- 1 is an even number.

A: For n=1 , 31-1= 3-1=2 , this is an even number Let for n=m, 3m-1 is an even number.  Assume that…

Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

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Q: 8. For any positive integer n, (i+1)·2' = n-2"+1. [PROOF BY MATHEMATICAL INDUCTION]

A: Case 1 : n=1 LHS = ∑i=11(i+1).(2i)= (1+1).(21)= (2)(2)= 4RHS = (1).(21+1)= (1)(22)= (1)(4)= 4 Hence…

Q: 1 Use Mathematical Induction to prove that 2 k2 k=1

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Q: 7. Use induction to prove Bernoulli's inequality: If 1 + x>0, then (1+ x)" al+ nx for all natural…

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Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: 9. Using a proof by induction, prove that for every natural number n the quantity 11n -6 is…

A: The statement is not true. Tale n=4

Q: Use mathematical induction to show that E"-, ar ark-1 a(1-r") for all integers n 2 1-r 1. (Assume r…

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Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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

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Q: (2) Prove by mathematical induction that for all integers n > 0, the number 42n+1 + (21 · 33"+1) is…

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Q: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

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Q: ) Prove every integer n >= 7, n! > 3n

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Q: b) Prove that 4" 10 by induction

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Q: Prove by induction that the sum of the first n integers starting at 1 (1,2,3,...n) is n(n+1)/2.

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Q: -ん 2 kodd

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Q: 16. Use mathematical induction to prove that, for all natural numbers n, n [n(n+1) i=1

A: Prnciple of mathrmatimat induction:Alogrithm              1. obtained  P(n) and understand its…

Q: # 2. Use mathematical induction II to prove that for all integers n 2 5, 1 + 4n < 2".

A: Given: n≥5 To prove: 1+4n&lt;2n using mathematical induction.

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: 2. Prove by induction (n+1) (n+2) - 1 is odd for all positive numbers of n >= 0

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Q: Prove by mathematical induction that the sum of the first n even positive integers is n' +(2n) = n^2…

A: to prove 2+4+6+..2n=n2+n by induction  Prove the result is true for n=1 LHS  LHS = 2 RHS…

Q: Prove the following equation for n^2 >5n+2 for every integer n>=100 by mathematical induction

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Q: Q1 Use mathematical induction to prove that (-5)"+1 – 1 3 + 3 * (-5) + 3 * (-5)² + 3 * (-5)³ + .… +…

A: Given equation is, 3+3(-5)+3(-5)2+3(-5)3+......+3(-5)n=(-5)n+1-1-2

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: 2 is a factor of n2 - n.Use mathematical induction to prove that the statement is true for every…

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Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: Prove using mathematical Induction: n 1 2. k=1

A: First we assume that result is true for n=k' then we prove that result is true for n=k'+1 .

Q: 1. Prove that, for any positive integer n," n´+2n is divisible by 3 " using the method of…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 2. Prove by induction that for all integers n > 0 5n+1 - 1 1+5+52 + ... + 5" 4

A: prove of the above by induction is given below...

Q: Prove by induction that for positive integers n:  4|(5^(n+1)+ 3^(2n-1)).

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Q: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 12 + 22 + ...+ n² = n(n+1)(2n + 1) 6

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Q: Use Mathematical Induction to prove that for all integers n > 1: (1 × 2) + (3 x 4) + (5 × 6) + · - -…

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Q: 1- Use mathema tical Induction to prove that 1+2 + --- 2. 4 (n+1)?/4 for all positive integers

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Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: Prove that for every non-negative integer n the value of n^2+n+41 is prime

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Q: 3. Prove that if n is odd, then 8 divides n2- 1. Prove that 6 divides n п. 4. Let n and k be two…

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Q: 1. Prove using mathematical induction that for all n E N. i. (4n3 – n) is divisible by 3. - ii. n<…

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Q: 1. Prove by mathematical induction, for any positive integer n, n³ + 2n is divisible by 3.

A: as per company guidelines we are supposed to answer only one question with maximum three subparts.…

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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Q: Prove, using mathematical induction that for an integer n 2 1, 1+5+9+... + (4n – 3) = n (2n – 1).

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Q: Prove by induction that the sum of the first n odd numbers is n². In other words, prove E1(2i – 1) =…

A: To prove by induction that the sum of the first natural numbers is n^2.

Q: 5. Use ordinary induction to prove that for every positive integer n, n°-n is a multiple of 6. Only…

A: To prove that  For every positive integer n, n^3-n is a multiple of 6 by using induction method.

Q: Prove by mathematical induction that 2n =0

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Q: Prove that n! > n2 for every integer n ≥ 4.

A: Use mathematical induction, to prove the statement n!&gt;n2 is true for all integer n≥4. Substitute…

Statements and Logic

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