For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

Q: n = n and n = 1. n Show that in

A: A factorial is the product of consecutive integers starting from 1. It is known that nr represents…

Q: (b)(i) Use Mathematical Induction to prove that for all positive integers n 21 1 1. 1 一 -十 +... 1x3…

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Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

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Q: For any k e Z, give a formula for the coefficient of x* in the expansion of (2-!)" 100 x²

A: Given expansion: x2-1x100 The given problem can be compared to the general form x+an. Here, n=100…

Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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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: 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: n is increasing, and those on w 3x + 5.

A: Introduction: The input values of a function f(x) where f(x) has minimum or maximum values are…

Q: 3. The fifth term in the expansion of (2x + k) is equal to the fourth term in the expansion of (2 +…

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

Q: Define the sequence co, C1, by the equations co = 0 and Cn = C\n/2 + 3 for all n > 0. Prove that ...…

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Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

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

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Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: Fix k, n E N with n > 2 and k <n – 2. Prove that: | )-(".) · ) (:) ) (^) = (":") 2 + k 2 n – 2 n n +…

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Q: B. Find the coefficients specified in each item. 2. a² in (6 - 1/2a)6 a4 4. k4 in (3k – 21)¹2

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Prove m+n-1 Σ k k k>0 k

A: We know that  nCr=n!r!(n-r)! When k= 0, the LHS becomes 0 When k=1, the LHS becomes…

Q: 2. Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n. for n ≥ 3, an = 2an-1+ bn-1…

A: NOTE:- AS POSTED MULTIPLE INDEPENDENT QUESTIONS. WE ARE ANSWERING ONLY FIRST QUESTIONS FOR YOU AS…

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

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: Find the value of K by determines. |1 2 3 0 1 2- |2 1 3 3 4 3 kl

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Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

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

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

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Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

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Q: The average pollen count in a certain city on day x of the pollen season is  P(x) = 4x − 0.2x2 for…

A: The average pollen count in a certain city on day x of the pollen season is P(x)=4x-0.2x2…

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: Prove that n² +n is divisible by 2, where n E N.

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Q: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: Find (a) an algebraic proof, and (b) a combinatorial proof for the following: Σ |2*3n-k k 4*. k k=0…

A: We have to find an algebraic proof, and combinatorial proof for ∑k=0nnk2k3n-k=∑k=0nnk4k  ⋯1.

Q: Show that Σ dn o(n) n for every positive integer n.

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Q: Q2) n(x) = x-1 , k(x) = (x – 1)/2, mx) = (x + 8)¯1 (x + 8)-1 Find a formula for ( m onok).

A: See answer below.

Q: In another proof, I found the following statement. Since we are given deg f(x) = n, then the…

A: Definition: (Quotient ring) Let R be a ring and I be an ideal in R. The quotient ring is defined as…

Q: In the expansion of (2r + 1)", the coefficient of the term in x is 40n, where n e positive integers.…

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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: Prove the following: for all integers n > 5, n2 < 2"

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: 3 and for n > 2, an = 2an-1+5, express an in terms of n. 1. Let, a1

A: The solution is given as

Q: 4 and for n > 3, an 2an-1 + an-2 + 5n, express an in terms of n %3D 2. Let, a1 = 3, a2

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Q: prove n2 + 5 greater than or equal to 4n for all integers n

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Q: B. Find the coefficients specified in each item. 2. a² in (6 — 1/1 a)6 4. k4 in (3k - 21) ¹2 12

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: 11) Which of the following shows the solution for n2- 2n < 24? 十十 A) 6-54-3-2-10123456 + +++++++…

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Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…