Tính :
(2025⁵⁰+2024⁵⁰)-(2026⁵⁰+2025⁵⁰)
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Khách
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HT
6
NL
Nguyễn Lê Phước Thịnh
CTVHS
7 giờ trước (15:57)
\(\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\cdots+\frac{1}{49\cdot50}\)
\(=1-\frac12+\frac13-\frac14+\cdots+\frac{1}{49}-\frac{1}{50}\)
\(=1+\frac12+\frac13+\cdots+\frac{1}{50}-2\left(\frac12+\frac14+\cdots+\frac{1}{50}\right)\)
\(=1+\frac12+\cdots+\frac{1}{50}-1-\frac12-\cdots-\frac{1}{25}\)
\(=\frac{1}{26}+\frac{1}{27}+\cdots+\frac{1}{50}\)
6 giờ trước (16:38)
\(VT=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+\cdots+\frac{1}{49.50}\)
\(=\left(1-\frac12\right)+\left(\frac13-\frac14\right)+\left(\frac15-\frac16\right)+\cdots+\left(\frac{1}{49}-\frac{1}{50}\right)\)
\(=\left(1+\frac13+\frac15+\cdots+\frac{1}{49}\right)-\left(\frac12+\frac14+\frac16+\cdots+\frac{1}{50}\right)\)
\(=\left(1+\frac12+\frac14+\frac14+\cdots+\frac{1}{50}\right)-2.\left(\frac12+\frac14+\frac16+\cdots+\frac{1}{50}\right)\)
\(=\left(1+\frac12+\frac13+\frac14+\cdots+\frac{1}{50}\right)-\left(1+\frac12+\frac13+\cdots+\frac{1}{25}\right)\)
\(=\frac{1}{26}+\frac{1}{27}+\cdots+\frac{1}{50}=VP\left(đpcm\right).\)
\(\left(2025^{50}+2024^{50}\right)-\left(2026^{50}+2025^{50}\right)\)
\(=2025^{50}+2024^{50}-2026^{50}-2025^{50}\)
\(=2024^{50}-2026^{50}+\left(2025^{50}-2025^{50}\right)\)
\(=2024^{50}-2026^{50}.\)
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