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For n € N, let n![denote the product 1-2-3-n and 0! = 1, and define (2)=; n! k!(n-k)! The binomial theorem asserts that i) Show ; ).…-+( 2 ). a²26² ++ | + ( ₁²₁ ) as² - ² + ( ^ ^ ) 8² 72 n-1 ab"-1 (a + b)² = ( ") a". + ( ₁² ) ₁²-¹b + ( for k = 0,1,...,n ~ ( ² ) + ( ₁ ² ₁ ) - ( " + ¹) = k k-1 + na a”¯¹b+ √n(n − 1)a”—²8² + ··· + nab”¯¹ + b² = } = Σ( ² ) ²8¹-² for k=1,2,...,n TL ii) Prove the binomial theorem using mathematical induction and part i).

For n € N, let n![denote the product 1-2-3-n and 0! = 1, and define (2)=; n! k!(n-k)! The binomial theorem asserts that i) Show ; ).…-+( 2 ). a²26² ++ | + ( ₁²₁ ) as² - ² + ( ^ ^ ) 8² 72 n-1 ab"-1 (a + b)² = ( ") a". + ( ₁² ) ₁²-¹b + ( for k = 0,1,...,n ~ ( ² ) + ( ₁ ² ₁ ) - ( " + ¹) = k k-1 + na a”¯¹b+ √n(n − 1)a”—²8² + ··· + nab”¯¹ + b² = } = Σ( ² ) ²8¹-² for k=1,2,...,n TL ii) Prove the binomial theorem using mathematical induction and part i).

Advanced Engineering Mathematics
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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For n € N, let n![denote the product 1-2-3-n and 0! = 1, and define (2)=; n! k!(n-k)! The binomial theorem asserts that (a + b)² = ( ") a". + ( ₁² ) ₁²-¹b + ( i) Show for k = 0,1,...,n ~ ( ² ) + ( ₁ ² ₁ ) - ( " + ¹) = k k-1 ; ).…-+( 2 ). a²26² ++ | + na a”¯¹b+ √n(n − 1)a”—²8² + ··· + nab”¯¹ + b² = } n-1 + ( ₂²1 ) ab²-¹ + ( " ) 8² - Σ( ² ) ¹²¹-* for k=1,2,...,n ii) Prove the binomial theorem using mathematical induction and part i).
Transcribed Image Text:For n € N, let n![denote the product 1-2-3-n and 0! = 1, and define (2)=; n! k!(n-k)! The binomial theorem asserts that (a + b)² = ( ") a". + ( ₁² ) ₁²-¹b + ( i) Show for k = 0,1,...,n ~ ( ² ) + ( ₁ ² ₁ ) - ( " + ¹) = k k-1 ; ).…-+( 2 ). a²26² ++ | + na a”¯¹b+ √n(n − 1)a”—²8² + ··· + nab”¯¹ + b² = } n-1 + ( ₂²1 ) ab²-¹ + ( " ) 8² - Σ( ² ) ¹²¹-* for k=1,2,...,n ii) Prove the binomial theorem using mathematical induction and part i).
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