Tìm số nguyên dương \(n\) bé nhất sao cho trong khai triển \((x+1)^{n}\) có hai hệ số liên tiếp nhau có tỷ số là \(\frac{7}{15}\) ( điè̀n đáp án vào ô trống).
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Ta có \((1+x)^{n}=C_{n}^{0}+C_{n}^{1} \cdot x+C_{n}^{2} \cdot x^{2}+\ldots+C_{n}^{n-1} \cdot x^{n-1}+C_{n}^{n} x^{n}\).
Số hạng thứ \(k\) và \(k+1\) theo khai triển trên là \(C_{n}^{k-1}, C_{n}^{k}\) với \(1 \leq k \leq n ; k, n \in \mathbb{N}\).
Theo giả thiết ta có:
\(\frac{C_{n}^{k-1}}{C_{n}^{k}}=\frac{7}{15} \Leftrightarrow \frac{k}{n-k+1}=\frac{7}{15} \Leftrightarrow 15 k=7(n-k+1) \Leftrightarrow 22 k=7(n+1) .\)
Do \((22 ; 7)=1\) nên \(n+1\) chia hết cho 22 . Vậy \(n=22 m-1, m \in \mathbb{N}\).
Vây số nguyên dương \(n\) bé nhất thỏa mãn đề bài là 21 .
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