HOTS Zone : Koefisien Binomial

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Berikut ini adalah kumpulan soal mengenai Koefisien Binomial. Jika ingin bertanya soal, silahkan gabung ke grup KLIK DI SINI.

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No. 1

Misalkan S = (x − 2)⁴ + 8(x − 2)³ + 24(x − 2)² + 32(x − 2) + 16. Apakah S jika dituliskan dalam sesedikit mungkin suku penjumlahan ?

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ALTERNATIF PENYELESAIAN

$\begin{array}{rl}S& =(x-2{)}^4+8(x-2{)}^3+24(x-2{)}^2+32(x-2)+16\\ & =(x-2{)}^4+4\cdot 2(x-2{)}^3+6\cdot 2^2(x-2{)}^2+4\cdot 2^3(x-2)+2^4\\ & =(x-2+2{)}^4\\ & =\text{color}blue\boxed{{\displaystyle \boxed{{\displaystyle \text{color}black{x}^4}}}}\end{array}$

Jadi, S jika dituliskan dalam sesedikit mungkin suku penjumlahan adalah x⁴.

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No. 2

Tentukan nilai $$\sum _{i=0}^{2011}(-1{)}^i(\genfrac{}{}{0ex}{}{2011}{i})$$

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ALTERNATIF PENYELESAIAN

Perhatikan bahwa $$(x+y{)}^n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})x^i{y}^{n-i}$$ Jika n = 2011, x = −1, y = 1, $$\begin{array}{rl}(-1+1{)}^{2011}& =\sum _{i=0}^{2011}(\genfrac{}{}{0ex}{}{2011}{i})(-1{)}^i(1{)}^{2011-i}\\ 0& =\sum _{i=0}^{2011}(-1{)}^i(\genfrac{}{}{0ex}{}{2011}{i})\\ \sum _{i=0}^{2011}(-1{)}^i(\genfrac{}{}{0ex}{}{2011}{i})& =\text{color}blue\boxed{{\displaystyle \boxed{{\displaystyle \text{color}black0}}}}\end{array}$$

Jadi, $\sum_{i=0}^{2011}(-1)^i\binom{2011}{i}=0$.

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