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Xét số hạng tổng quát:
\(k^4+\frac{1}{4}=\left(k^4+2\cdot\frac{1}{2}\cdot k^2+\frac{1}{4}\right)-k^2\)=\(\left(k^2+\frac{1}{2}\right)^2-k^2\)
= \(\left(k^2+\frac{1}{2}-k\right)\left(k^2+\frac{1}{2}+k\right)\)
Thay k từ 1 đến 12 ta được:
A=\(\frac{\frac{1}{2}\cdot\left(2+\frac{1}{2}\right)\left(6+\frac{1}{2}\right)\left(12+\frac{1}{2}\right)...\left(110+\frac{1}{2}\right)\left(132+\frac{1}{2}\right)}{\left(2+\frac{1}{2}\right)\left(6+\frac{1}{2}\right)...\left(132+\frac{1}{2}\right)\left(152+\frac{1}{2}\right)}\)=\(\frac{\frac{1}{2}}{152+\frac{1}{2}}=\frac{1}{305}\)
\(A=\frac{\left(1+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right).....\left(51^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)....\left(52^4+\frac{1}{4}\right)}\)
\(=\frac{\left(1+1+\frac{1}{2}\right)\left(1-1+\frac{1}{2}\right)....\left(11^2-11+\frac{1}{2}\right)}{\left(2+2^2+\frac{1}{2}\right)\left(2^2-2+\frac{1}{2}\right)....\left(12^2-12+\frac{1}{2}\right)}\)
\(=\frac{\frac{1}{2}\left(1.2+\frac{1}{2}\right)\left(2.3+\frac{1}{2}\right)....\left(11.12+\frac{1}{2}\right)}{\left(2.3+\frac{1}{2}\right)\left(3.4+\frac{1}{2}\right)....\left(12.13+\frac{1}{2}\right)}\)
\(=\frac{\frac{1}{2}}{12.13+\frac{1}{2}}\)
\(=\frac{1}{313}\)
Chúc bạn học tốt !!!