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).

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