an+1 - a 1.) Let a +1 be a real number. Prove that a + a² + a³ + • · ·+ a" for all integers n> 1. a – 1

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

Q: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

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

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Q: 4. Prove by mathematical induction that for every positive integer n, II (G- 1 %3D 3i-1 31 3i+1…

A: We have to prove given expression using mathematical induction.

Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

Q: Use a direct proof. Let "a", "b", and "c" be integers, where "a" is not equal to 0. “if a|b and b|c,…

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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: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

Q: 1- gn+1 Σ %3D :- 1-τ i=0

A: Solution

Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove that there is a positive number z so that z^5 + 10z = 10.

A: TO PROVE:-  There is a positive number z satisfying z5+10z=10 GIVEN :- z5+10z=10

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

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Q: a. Prove that the v63 iş irrational b. For all integers, prove that 4 (n2 + 2) c. Prove or disprove:…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

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

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Q: 4- Let n be a positive integer, and define T 2 -1. Prove that T, has at least n different divisors.…

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Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.…

Q: a Positive prove that if is in integer then 133 divides Itull 2n-) ?n-1 + 12

A: We will prove the given statement.

Q: Prove that the statement is true for every positive integer n.

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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Q: Prove by Induction that if                                          a1,a2,a3,.........an belongs to…

A: I have proved for n=1, 2 and by assuming n=k i have proved for n=k+1

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

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

Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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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: 9. (a) Show that if n is an integer, then: -o3* (") = 4". %3D (b) Let n be a positive integer. Show…

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Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n. 22+42+62+...+2n2=2n2n+1n+13 We prove…

Q: Prove using Direct Proof: For all integers a,b and c if a|b and b|c then a|c

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: Prove that 1 + 3 + • • • + (2n - 1) = n2 for all natural numbers n.

A: Our Aim is to prove that 1 + 3 + .... + (2n - 1) = n2 for all natural numbers n.

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: part ii and iii, thank you

A: According to pascal law,

Q: let n be a positive integer not exceeding 20. How many values of n such that the equation x^2 +…

A: We have to find the values of n for which x^2+y^3n+1=1.

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

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Q: Prove that n^2+5 >=4n for all integers, n>0.

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

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Q: Let n>=2 and k be any integer, then prove that (n-1)^2 divides (n^k )-1 iff n-1 divides k.

A: See the attachment

Q: 2n° 1. Prove that for all positive integers n > 1, the inequality 2" < <(") holds. п

A: Consider the given statement be P(n) as; Apply induction methods to prove the above statement P(n)…

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: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Prove that A,(3, 2) = q² for any integer q>2. %3D

A: To prove : Aq3, 2=q2 for any integer q≥2 Pre-requisite : 1. Sphere-covering bound : For an integer…

Q: 2- 3" – 1. 3an + 2, for all nonnegative integers n. Prove that an %3D Let ao 1, and let an+! %3D

A: Given a0 =1 an+1 = 3an +2 for all non negative integers n. We need to prove an = 2.3n -1

Q: Prove that each statement is true for all natural numbers. 1) 3+7+11+...(4n-1)=2n^2+n

A: Since you have posted multiple questions i can do first question as per our company guidelines…