(a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficientand use it to prove the property that generates each row of Pascal's triangle fromthe previous one, namely(1),n( 7 ) + ( x + 1)- (2+1)kfor all n, k in the appropriate range.(b) Let F := F(X,Y) (Xª – 2Y)º. Use the Binomial Theorem to determine thecoefficients with which the following terms appear in F:(i) X ³²Y;(ii) X36Y:(iii) X³y7.=

a) Given: n,k∈ℤ with 0≤k≤n To prove that nk+nk+1=n+1k+1

(a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficient (7) and use it to prove the property that generates each row of Pascal's triangle from the previous one, namely n+1 (1) + (x + 1) = (+¹) k+1 for all n, k in the appropriate range. = (b) Let F = F(X,Y) (X4 – 2Y)⁹. Use the Binomial Theorem to determine the coefficients with which the following terms appear in F: (i) X ³²Y; (ii) X³6Y; (iii) X³y7.

(a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficient (7) and use it to prove the property that generates each row of Pascal's triangle from the previous one, namely n+1 (1) + (x + 1) = (+¹) k+1 for all n, k in the appropriate range. = (b) Let F = F(X,Y) (X4 – 2Y)⁹. Use the Binomial Theorem to determine the coefficients with which the following terms appear in F: (i) X ³²Y; (ii) X³6Y; (iii) X³y7.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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