(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Let n, k € Z with 0 ≤ k ≤n. Write down the formula for the binomial coefficient
and use it to prove the property that generates each row of Pascal's triangle from
the previous one, namely
(1),
n
( 7 ) + ( x + 1)
- (2+1)
k
for all n, k in the appropriate range.
(b) Let F := F(X,Y) (Xª – 2Y)º. Use the Binomial Theorem to determine the
coefficients with which the following terms appear in F:
(i) X ³²Y;
(ii) X36Y:
(iii) X³y7.
=
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