Q6. Let n be n₁ + N₂ + N3 +・・・+nt, where each n; is a positive integer. Use the definition of binomial coefficients to prove that n₁ + n1 + nt -1₁) (1²₂ + nt ) (ns + ... + ₁) ... (1²) (1² = n3 n₂ + ... + nt n2 n! nt n₁!n₂!...nt!

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: Show that 1+2+3+...+n= ("""). (";"').

A:

Q: (x – 2y2)5 ·+ bx³y* + •…· | ||

A: It is an important part of statistics . It is widely used .

Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € N.

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Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

A:

Q: Write the binomial expansion for (x+ 2y)*. (x + 2y)* =D

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Q: Q1 (b): If Xi= 1,3,5,7,9,11 and Yi= 0,2,4,6,8,10. The values of the :following term is i=n 2n – 3…

A: Given,∑i=1i=n2n-3πyi-11+xii-1here n=6

Q: How many strings of 1our decimal digits have all four digits that are different? How many strings of…

A:

Q: Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1)…

A: Given, r is a real number and k is an integer  The Binomial Coefficient rk by rk= r (r - 1) --- (r -…

Q: 6. Use the binomial theorem to prove that 3 = C(n,0) + 2C(n,1)+2²C(n, 2) + +2"C(n, n)

A: By using the binomial thorem to prove that, 3n=Cn, 0+2Cn, 1+22Cn, 2+...+2nCn, n

Q: Prove that the sum 2 + 22 + 23 + 24 + · . . + 22019 + 22020 is divisible by 30.

A: The sum of the geometric series a, ar, ar2, ar3,…, arn-1 is given as the following formula.…

Q: Use the binomial theorem to expand the following expression. (u – v)5 By the binomial theorem with 5…

A: Given expression (u-v)5 Using binomial theorem,

Q: Using the binomial coefficients (2) = n n! (n-k)!k! (1) The sum first n integers: 1 + 2 + ... + n =…

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Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: Find a Big-O estimate of (2n+n2)(n3+3n)

A: Given the given expression is 2n+n2n3+3n. Big-O notation: the fn is Ogn for positive constant C and…

Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

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Q: Prove that n>= 12 is a sum of two composite number.|

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Q: expand each expression using the Binomial Theorem

A: Given: 19.(x-2)6

Q: Use Pascal's Triangle to expand the binomial (x + 1)° O 25 + 5x4 + 10x + 10x2 + 5x + 1 O g5 - 5x1-…

A: Pascal's triangle…

Q: Let denote the binomial coefficient (also written as C(n, r) or nCr). The sum of (C) + (4) +…

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Q: Which of the following is the value of the coefficient y on the right side of the equation given…

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Q: п n 9. Using the Binomial Theorem, show that Σ = n 2"-1. k k=0

A: We need to show that- ∑k=0nknk = n2n-1

Q: Σ >k· k! = (n + 1)! – 1 k=1 for every positive integer, n

A: We will prove this by principle of mathematical induction.

Q: Write the recursive definition for an = pn n! where a1 = p.

A: The recursive formula is used to describe the relationship between two terms of any sequence. The…

Q: (2x-3y) =

A:

Q: which pair of binomials disproves Eric's ㅇ (z-2)(z + 3) ㅇ (z-2) (z + 2) z+3)(3 2)

A: Solution: