Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1) (r-k+1) O-F k! if k 21 if k=0 if k ≤ -1. 1. For any real number r and any integer r #k, show that a) ()=(²¹), and b) () = (-1)k (r+k-1). Hint: use the definition of () from Page 137.

Q: Find the solution to this recursive relation an = an-1+3, a0=1

A: To Find the solution to this recursive relation an = an-1+3, a0=1

Q: 1+ x Σ Aa" (1 – 2)? n=0

A:

Q: The coefficient of x²y4z5 in the expansion of (3x+y+z)¹¹ is

A: The given expression is: (3x + y + z)11 To find: The coefficient of x2y4z5 in the expansion of (3x +…

Q: Find a given that ||a i+ (a – 1)j + (a + 1)k|| = 7.

A: Given:

Q: Suppose X ÷ ( – 2) is 7x? – 2x – 6 with remainder 5. - - Then X %3D

A:

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

A:

Q: (c) Find an element h of S10 such that hog (1 8 10 2 7 3 6 4), = and write it in two-line notation.…

A: (c) In one-line notation, g can be written as        g = (1  6  9  3  10  8  7)(4  5)    Then,…

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

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

A: The binomial expansion of (a+b)^n is

Q: Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For…

A:

Q: #2. The binomial theorem states that for any real numbers a and b, In (a + b)n = =Σ" ("a²-kbk for…

A:

Q: Use the Geometric Sum identity to prove that, if r 70, then (1 – x)3 1 =1+ (1 - x) + (1 – x)² + |

A: The objectives to use geometry sum identity to prove the following expression.

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: 1.5 Inx - In(x2 - 1) + 2ln(x + 3)

A: Logarithm simplification

Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

A:

Q: For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive…

A: Let's break down the problem into two parts: (a) For every natural number 𝑥, there is a natural…

Q: Given (xi , xi2022 i ) ∀i = 0, ..., n, where x0 = −1, x1 = 0, x2 = 1, x3 = 2, ..., xn = n − 1. Write…

A: Given xi,xi2022∀i=0,1,2,...,n  Where x0=-1, x1=0, x2=1, x3=2,..., xn=n-1