Cho dãy số \(\left( {{u}_{n}} \right)\) biết \({{u}_{n}}={{n}^{2}}+2n,\,n\in {{\mathbb{N}}^{\text{*}}}\).
a) Số hạng đầu tiên của dãy số là \({{u}_{1}}=3\).
b) Dãy số \(\left( {{u}_{n}} \right)\) là một dãy số giảm.
c) Số \(143\) là số hạng thứ \(13\) trong dãy số \(\left( {{u}_{n}} \right)\).
d) \(\forall n\in {{\mathbb{N}}^{\text{*}}}\) thì \(\frac{1}{{{u}_{1}}}+\frac{1}{{{u}_{2}}}+\frac{1}{{{u}_{3}}}+...+\frac{1}{{{u}_{n}}}=\frac{3{{n}^{2}}+5n}{2\left( n+1 \right)\left( n+2 \right)}\).
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
a) Ta có \({{u}_{1}}={{1}^{2}}+2.1=3\)
b) \(\forall n\in {{\mathbb{N}}^{\text{*}}}:\,{{u}_{n+1}}-{{u}_{n}}={{\left( n+1 \right)}^{2}}+2\left( n+1 \right)-\left( {{n}^{2}}+2n \right)=2n+3>0\).
Suy ra dãy số \(\left( {{u}_{n}} \right)\) là một dãy số tăng.
c) Ta có \({{u}_{n}}=143\Leftrightarrow {{n}^{2}}+2n=143\Leftrightarrow \left[ \begin{align} n=11 \\n=-13 \\ \end{align} \right.\).
Do \(n\in {{\mathbb{N}}^{\text{*}}}\) nên \(n=11\), nghĩa là số \(143\) là số hạng thứ \(11\).
d) Ta có \({{u}_{n}}={{n}^{2}}+2n=n\left( n+2 \right)\)
Do đó \(\frac{1}{{{u}_{1}}}+\frac{1}{{{u}_{2}}}+\frac{1}{{{u}_{3}}}+...+\frac{1}{{{u}_{n}}}\)\(=\frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+...+\frac{1}{n\left( n+2 \right)}\)
Phân tích: \(\frac{1}{1.3}=\frac{1}{2}\left( \frac{1}{1}-\frac{1}{3} \right);\)\(\frac{1}{2.4}=\frac{1}{2}\left( \frac{1}{2}-\frac{1}{4} \right);\)...
\(\frac{1}{n.\left( n+2 \right)}=\frac{1}{2}\left( \frac{1}{n}-\frac{1}{n+2} \right)\).
Ta được:
\(\begin{align}\frac{1}{{{u}_{1}}}+\frac{1}{{{u}_{2}}}+\frac{1}{{{u}_{3}}}+...+\frac{1}{{{u}_{n}}}=\frac{1}{2}\left( \frac{1}{1}+\frac{1}{2}-\frac{1}{n+1}-\frac{1}{n+2} \right) =\frac{1}{2}.\frac{3\left( n+1 \right)\left( n+2 \right)-2\left( n+2+n+1 \right)}{2\left( n+1 \right)\left( n+2 \right)} =\frac{3{{n}^{2}}+5n}{4\left( n+1 \right)\left( n+2 \right)} \\ \end{align}\)
Bộ Đề Kiểm Tra Giữa Học Kì I - Toán 11 - Kết Nối Tri Thức Với Cuộc Sống được thiết kế chuyên sâu, tập trung vào các kiến thức nền tảng như hàm số lượng giác, phương trình lượng giác phức tạp, dãy số, cấp số cộng và cấp số nhân, cùng với các bài toán tổ hợp và xác suất mang tính ứng dụng cao. Với hệ thống câu hỏi được sắp xếp từ cơ bản đến nâng cao, đề thi không chỉ giúp học sinh nắm vững cách giải quyết các bài toán tiêu chuẩn mà còn nâng cao khả năng phân tích và tư duy toán học một cách logic và sáng tạo. Đây là tài liệu hữu hiệu để học sinh làm quen với các dạng bài tập phức tạp, rèn luyện kỹ năng xử lý tình huống toán học linh hoạt, chuẩn bị sẵn sàng cho kỳ thi với kết quả tốt nhất.
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