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Câu 24 trang 115 SGK Đại số và Giải tích 11 Nâng cao

Cho cấp số cộng (un)
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Đề bài

Cho cấp số cộng (un♐) với công sai d và cho các số nguyên dương m và k, với \(m ≥ k\). Chứng minh rằng \({u_m} = {u_k} + \left( {m-k} \right)d\).

Áp dụng : Hãy tìm công sai d của cấp số cộng (un) mà \({u_{18}} - {u_3} = 75\).

Phương pháp giải - Xem chi tiết

Viết công thức tính \(u_m,u_k\) theo \(u_1,d\) rồi trừ hai số hạng cho nhau suy ra ĐPCM. Sử dụng công thức \[{u_n} = {u_1} + \left( {n - 1} \right)d\]

Lời giải chi tiết

Ta có:

\(\eqalign{
& {u_m} = {u_1} + \left( {m - 1} \right)d\,\left( 1 \right) \cr 
♚& {u_k} = {u_1} + \left( {k - 1} \right)d\,\left( 2 \right) \cr} \)

Lấy (1) trừ (2) ta được :\({u_m} - {u_k} \)\( = {u_1} + \left( {m - 1} \right)d - {u_1} - \left( {k - 1} \right)d \)\(= \left( {m - 1 - k + 1} \right)d\)\(= \left( {m - k} \right)d\)\(\Rightarrow {u_m} = {u_k} + \left( {m - k} \right)d\)Áp dụng :Ta có:

\(\eqalign{
& {u_{18}} - {u_3} = \left( {18 - 3} \right)d = 15d = 75 \cr 
& \Rightarrow d = 5 \cr} \)

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