b) P= 3 7.5 + 3 7.5^ 2 + 3 7.5^ 3 +...+ 3 7.5^ 2006 giúp mình với ạ
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Huy
P=37.5+37.52+37.53+...
=> P = 37 . (5+5^2+5^3+...+5^{2006})
M = 5+5^2+5^3+...+5^{2006}
=>5M = 5^2+5^3+5^4+...+5^{2007}
=> 5M - M = (5^2+5^3+5^4+...+5^{2007}) - (5 + 37.5^2 + 37.5^3 + ... + 37.5^{2006} )
=> 4M = 5^{2007} - 5
=> M= \frac{5^{2007}-5}{4}
=> P=37 . \frac{5^{2007}-5}{4} = \frac{37.(5^{2007}-5)}{4}
Đáp án:
P=\dfrac{21.5^{2007}-105}4.
Giải thích các bước giải:
\begin{array}{l}P=3.7.5+3.7.5^2+3.7.5^3\ +\,.\!.\!.+\ 3.7.5^{2006}\\\Rightarrow P=3.7.(5+5^2+5^3\ +\,.\!.\!.+\ 5^{2006})\\\Rightarrow\dfrac P{3.7}=5+5^2+5^3\ +\,.\!.\!.+\ 5^{2006}\\\Rightarrow\dfrac{5P}{21}=5.(5+5^2+5^3\ +\,.\!.\!.+\ 5^{2006})\\\Rightarrow \dfrac{5P}{21}=5^2+5^3+5^4\ +\,.\!.\!.+\ 5^{2007}\\\Rightarrow\dfrac{5P}{21}-\dfrac P{21}=(5^2+5^3+5^4\ +\,.\!.\!.+\ 5^{2007})-(5+5^2+5^3\ +\,.\!.\!.+\ 5^{2006})\end{array}\\\Rightarrow\dfrac{4P}{21}=5^{2007}-5\\\Rightarrow \dfrac P{21}=\dfrac{5^{2007}-5}4\\\Rightarrow P=\dfrac{21.(5^{2007}-5)}4\\\Rightarrow P=\dfrac{21.5^{2007}-105}4
Vậy P=\dfrac{21.5^{2007}-105}4.
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