(a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer (positive, negative or zero).

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

A: Sum of two integers of the form 4k+1

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

A:

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: Prove the bi-conditional: prove that if 5n+2 is even, then n is even, where n is an integer.

A:

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: 2. Let n be a positive integer. Show that 4n 4n-k (4n 3 4n k=2n (Hint: an argument similar to the…

A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

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

A:

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: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: Prove that for any nonnegative integer n. n + +.. + 22m i+1 i +2 n+1 i=0 the

A:

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: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

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

A:

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

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

A:

Q: 10. Prove that the product of two integers is odd if and only if both of the integers are odd. 11.…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Can you show me step by step how to do a proof by contradiction for the theorem that if the result…

A:

Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

A:

Q: 2"-1 for positive integers n. 2n

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

A:

Q: r(r+1) n+1 8. Prove by mathematical induction that 5"-4n - 1 is divisible by 16, for all positive…

A: As per our guidelines we are supposed to solve only one answer . Please repost another question as…

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: se indirect proof to show that, "If n is an integer and 3n + 2 is even, then n is even

A: Let us solve the classical algebra problem in the next step.

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

A:

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

A:

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: The binomial theorem states that for any real numbers a and b, (a +b)" = E o (") a"-* for any…

A:

Q: 1. Prove that 3 + 3 x 5 + 3 x 52 + 3 x 53 + ..+ 3 x 5° = 3(5n+1 – 1)/4 whenever n is a nonnegative…

A:

Q: 1+ 3k 1- 3k] 1 – 3k 1+3k 2 - prove that AK: 2 1 For A : -1 -

A: Given, A=2-1-12   ...i    ....(i)  We need to show that Ak=121+3k1+3k1+3k1+3k    ...ii     Now, A…

Q: Prove that (f^2)_1 + (f^2)_2 + ⋯ + (f^2)_n = f_n*f_(n+1) when n is a positive integer.

A: We have to show that  f12+f22+f32+.....fn2=fnfn+1 where n is a positive integer and fn denotes the…

Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers is given by the…

A: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers is given bythe…

Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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

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Q: Show for all integers n2 0 that 91 (4" - 1). a.

A: A. We will solve the question using the method of mathematical induction.

Q: Use mathematical induction to prove the inequalities. Prove that n2 - 7n + 12 is nonnegative…

A:

Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: Prove that 1- 1! +2 · 2! + .. +n ·n! = (n+ 1)! – 1 whenever n is a positive integer.

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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 that the square of any odd integer n, (n =4q +3 for some integer q) has the form 8m + 1 for…

A: Here we have to prove that any odd integer n which is of the form (n = 4q + 3) , where q is an…

Q: a) By using mathematical induction, prove that the given equation is true for all positive integers.…

A: I will Solve (a) For you. Given, 2+6+10+.......+(4n-2)=2n2 We have to prove this equation is true…

Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

A:

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.