Tính tích sau rồi tìm số nghịch đảo của kết quả:
\(T = \left( {1 – {1 \over 3}} \right).\left( {1 – {1 \over 5}} \right).\left( {1 – {1 \over 7}} \right).\left( {1 – {1 \over 9}} \right).\left( {1 – {1 \over {11}}} \right)\left( {1 – {1 \over 2}} \right).\left( {1 – {1 \over 4}} \right).\left( {1 – {1 \over 6}} \right).\left( {1 – {1 \over 8}} \right).\left( {1 – {1 \over {10}}} \right)\)
Giải
\(T = \left( {1 – {1 \over 3}} \right).\left( {1 – {1 \over 5}} \right).\left( {1 – {1 \over 7}} \right).\left( {1 – {1 \over 9}} \right).\left( {1 – {1 \over {11}}} \right)\left( {1 – {1 \over 2}} \right).\left( {1 – {1 \over 4}} \right).\left( {1 – {1 \over 6}} \right).\left( {1 – {1 \over 8}} \right).\left( {1 – {1 \over {10}}} \right)\)
\(\eqalign{
& = {2 \over 3}.{4 \over 5}.{6 \over 7}.{8 \over 9}.{{10} \over {11}}.{1 \over 2}.{3 \over 4}.{5 \over 6}.{7 \over 8}.{9 \over {10}} \cr
& = \left( {{2 \over 3}.{1 \over 2}} \right).\left( {{4 \over 5}.{3 \over 4}} \right).\left( {{6 \over 7}.{5 \over 6}} \right).\left( {{8 \over 9}.{7 \over 8}} \right).{9 \over {10}}.{{10} \over {11}} \cr
& = \left( {{2 \over 3}.{1 \over 2}} \right).\left( {{4 \over 5}.{3 \over 4}} \right).\left( {{6 \over 7}.{5 \over 6}} \right).\left( {{8 \over 9}.{7 \over 8}} \right).{9 \over {10}}.{{10} \over {11}} \cr
& = {1 \over 3}.{3 \over 5}.{5 \over 7}.{7 \over 9}.{9 \over {10}}.{{10} \over {11}} \cr
& = {1 \over {11}} \cr} \)
\(T = {1 \over {11}}\) có số nghịch đảo là 11