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Observe the characteristics of the following monomials: 1 / 2x & # 178; y, - 1 / 4x & # 178; Y & # 178;, 1 / 8x & # 178; Y & # 179;, - 1 / 16x & # 178; y Λ 4 (Λ table) Observe the characteristics of the following monomials: 1 / 2x-178; y, - 1 / 4x-178; y-178;, 1 / 8x-178; y-179;, - 1 / 16x-178; y-4 (Λ denotes power) . (1) Please write the eighth monomial according to this rule. How many times is it? (2) Try to guess: what is the nth monomial? What are its coefficients? How many times?
1. The eighth monomial: 1 / 2 * (- 1 / 2) ^ (8-1) x & # 178; y ^ 8 = - 1 / 256 x & # 178; y ^ 8.10 times (sum of X and y)
2. The nth monomial: 1 / 2 * (- 1 / 2) ^ (n-1) x & # 178; y ^ n. The coefficient is 1 / 2 * (- 1 / 2) ^ (n-1). Times: n + 2
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