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Câu 46 trang 123 SGK Đại số và Giải tích 11 Nâng cao

Cho các dãy số (un)
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Cho các dãy số (un) và (vn🍬) với  \({u_n} = {{{n^2} + 1} \over {n + 1}}\text{ và }{v_n} = {{2n} \over {n + 1}}\)

a) Hãy xác định số hạng tổng quát của dãy số (an) với an = un + vn

b) Hãy xác định số hạng tổng quát của dãy số (bn) với bn = un – vn

c) Hãy xác định số hạng tổng quát của dãy số (cn) với cn = un.vn

d) Hãy xác định số hạng tổng quát của dãy số (dn) với  \({d_n} = {{{u_n}} \over {{v_n}}}\)

Chú ý

Các dãy số (an), (bn), (cn), (dn) nêu trên thường được kí hiệu tương ứng bởi (un + vn), (un – vn), (un.vn),\(\left( {{{{u_n}} \over {{v_n}}}} \right)\).

LG a

Hãy xác định số hạng tổng quát của dãy số (an) với an = un + vn

Lời giải chi tiết:

Ta có: \({a_n} = {u_n} + {v_n} = {{{n^2} + 1} \over {n + 1}} + {{2n} \over {n + 1}} \) \( = \frac{{{n^2} + 2n + 1}}{{n + 1}}\) \(= {{{{\left( {n + 1} \right)}^2}} \over {n + 1}} = n + 1\)

LG b

Hãy xác định số hạng tổng quát của dãy số (bn) với bn = un – vn

Lời giải chi tiết:

Ta có: \({b_n} = {u_n} - {v_n} = {{{n^2} + 1} \over {n + 1}} - {{2n} \over {n + 1}}\) \( = \frac{{{n^2} - 2n + 1}}{{n + 1}}= {{{{\left( {n - 1} \right)}^2}} \over {n + 1}}\)

LG c

Hãy xác định số hạng tổng quát của dãy số (cn) với cn = un.vn

Lời giải chi tiết:

Ta có: \({c_n} = {u_n}{v_n} = \frac{{{n^2} + 1}}{{n + 1}}.\frac{{2n}}{{n + 1}}= {{2n\left( {{n^2} + 1} \right)} \over {{{\left( {n + 1} \right)}^2}}}\)

LG d

Hãy xác định số hạng tổng quát của dãy số (dn) với  \({d_n} = {{{u_n}} \over {{v_n}}}\)

Lời giải chi tiết:

Ta có: \({d_n} = {{{u_n}} \over {{v_n}}}  = \frac{{{n^2} + 1}}{{n + 1}}:\frac{{2n}}{{n + 1}}\) \(= \frac{{{n^2} + 1}}{{n + 1}}.\frac{{n + 1}}{{2n}}= {{{n^2} + 1} \over {2n}}\)

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