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Giải mục 1 trang 52, 53 SGK Toán 11 tập 1 - Kết nối tri thức

Cho dãy số (left( {{u_n}} right)) với ({u_n} = {3.2^n}) a) Viết năm số hạng đầu của dãy số này b) Dự đoán hệ thức truy hồi liên hệ giữa ({u_n}) và ({u_{n - 1}})

♊Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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HĐ 1

Video hướng dẫn giải

Cho dãy số \(\left( {{u_n}} \right)\) với \({u_n} = {3.2^n}\) a) Viết năm số hạng đầu của dãy số này. b) Dự đoán hệ thức truy hồi liên hệ giữa \({u_n}\) và \({u_{n - 1}}\).

Phương pháp giải:

Thay n tương ứng vào công thức số hạng tổng quát \({u_n}\). Xét tỷ số \(\frac{{{u_n}}}{{{u_{n - 1}}}}\) để tìm mối liên hệ giữa \({u_n}\) và \({u_{n - 1}}\).

Lời giải chi tiết:

a) Ta có: \({u_1} = 6,\;\;\;\;{u_2} = 12,\;\;\;\;\;{u_3} = 24,\;\;\;\;\;{u_4} = 48,\;\;\;\;\;{u_5} = 96\). b) Hệ thức truy hồi liên hệ giữa \({u_n}\) và \({u_{n - 1}}\) là: \({u_n} = 2{u_{n - 1}}\).

CH 1

Video hướng dẫn giải

Dãy số không đổi a,a, a,... có phải là một cấp số nhân không?

Phương pháp giải:

Để chứng minh dãy số \(\left( {{u_n}} \right)\) gồm các số khác 0 là một cấp số nhân, hãy chứng minh tỷ số \(\frac{{{u_n}}}{{{u_{n - 1}}}}\) = q không đổi.

Lời giải chi tiết:

Ta thấy tỉ số của các số hạng là \(\frac{a}{a} = 1, \forall n \ge 2\). Như vậy, dãy số không đổi a,a, a,... là một cấp số nhân.

LT 1

Video hướng dẫn giải

Cho dãy số \({u_n}\)với \({u_n} = {2.5^n}\). Chứng minh rằng dãy số này là một cấp số nhân. Xác định số hạng đầu và công bội của nó.

Phương pháp giải:

Để chứng minh dãy số \(\left( {{u_n}} \right)\) gồm các số khác 0 là một cấp số nhân, hãy chứng minh tỷ số \(\frac{{{u_n}}}{{{u_{n - 1}}}}\) = q không đổi.

Lời giải chi tiết:

Ta có: \(\frac{{{u_n}}}{{{u_{n - 1}}}} = \frac{{2 \times {5^n}}}{{2 \times {5^{n - 1}}}} = \frac{{2 \times {5^n}}}{{2 \times {5^{n}.5^{- 1}}}} = 5,\;\forall n \ge 2\). Vậy dãy số \(\left( {{u_n}} \right)\) là một cấp số nhân với \({u_1} = 10\) và công bội \(q = 5\).

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