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Giải bài 3 trang 60 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Cho cấp số cộng \(\left( {{u_n}} \right)\) có số hạng tổng quát: \({u_n} = 7n - 3\). a) Tìm số hạng đầu và công sai của cấp số cộng \(\left( {{u_n}} \right)\). b) Tìm \({u_{2012}}\). c) Tính tổng của 100 số hạng đầu tiên của cấp số cộng \(\left( {{u_n}} \right)\). d) Số 1 208 là số hạng thứ bao nhiêu của cấp số cộng \(\left( {{u_n}} \right)\)?

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Đề bài

Cho cấp số cộng \(\left( {{u_n}} \right)\) có số hạng tổng quát: \({u_n} = 7n - 3\). a) Tìm số hạng đầu và công sai của cấp số cộng \(\left( {{u_n}} \right)\). b) Tìm \({u_{2012}}\). c) Tính tổng của 100 số hạng đầu tiên của cấp số cộng \(\left( {{u_n}} \right)\). d) Số 1 208 là số hạng thứ bao nhiêu của cấp số cộng \(\left( {{u_n}} \right)\)?

Phương pháp giải - Xem chi tiết

a) Sử dụng kiến thức về khái niệm cấp số cộng để tính: Cấp số cộng là một dãy số (vô hạn hoặc hữu hạn) mà trong đó, kể từ số hạng thứ hai, mỗi số hạng đều bằng tổng của số hạng đứng ngay trước nó với một số d không đổi, nghĩa là: \({u_{n + 1}} = {u_n} + d\) với \(n \in \mathbb{N}*\). Số d được gọi là công sai của cấp số cộng. b, d) Sử dụng kiến thức về số hạng tổng quát của cấp số cộng để tính: Nếu một cấp số cộng \(\left( {{u_n}} \right)\) có số hạng đầu \({u_1}\) và công sai d thì số hạng tổng quát \({u_n}\) của nó được xác định bởi công thức: \({u_n} = {u_1} + \left( {n - 1} \right)d,n \ge 2\). c) Sử dụng kiến thức về tổng của n số hạng đầu tiên của cấp số cộng để tính: Nếu một cấp số cộng \(\left( {{u_n}} \right)\) có số hạng đầu \({u_1}\) và công sai d. Đặt \({S_n} = {u_1} + {u_2} + ... + {u_n}\), khi đó \({S_n} = \frac{{n\left( {{u_1} + {u_n}} \right)}}{2}\) hay \({S_n} = \frac{{n\left[ {2{u_1} + \left( {n - 1} \right)d} \right]}}{2}\).

Lời giải chi tiết

a) Số hạng đầu của cấp số cộng là: \({u_1} = 7.1 - 3 = 4\). Lại có: \({u_2} = 7.2 - 3 = 11\). Do đó, \(d = {u_2} - {u_1} = 11 - 4 = 7\). Vậy công sai của cấp số cộng là \(d = 7\). b) Ta có: \({u_{2012}} = 7.2012 - 3 = 14\;081\). c) Tổng của 100 số hạng đầu tiên của cấp số cộng là: \({S_{100}} = \frac{{100\left[ {2.4 + \left( {100 - 1} \right).7} \right]}}{2} = 35\;050\) d) Ta có: \(1\;208 = 7n - 3 \Leftrightarrow 7n = 1\;211 \Leftrightarrow n = 173\) Do đó, số 1 208 là số hạng thứ 173 của cấp số cộng \(\left( {{u_n}} \right)\).

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