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Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\dfrac{a_1}{a_2}=\dfrac{a_2}{a_3}=\dfrac{a_3}{a_4}=....=\dfrac{a_{2000}}{a_{2001}}=\dfrac{a_1+a_2+a_3+....+a_{2000}}{a_2+a_3+a_4+....+a_{2001}}\)
\(\Rightarrow\dfrac{a_1}{a_2}.\dfrac{a_2}{a_3}.\dfrac{a_3}{a_4}......\dfrac{a_{2000}}{a_{2001}}=\left(\dfrac{a_1+a_2+a_3+....+a_{2000}}{a_2+a_3+a_4+....+a_{2001}}\right)^{2000}\)
\(\Rightarrow\dfrac{a_1}{a_{2001}}=\left(\dfrac{a_1+a_2+a_3+....+a_{2000}}{a_2+a_3+a_4+....+a_{2001}}\right)^{2000}\)(đpcm)
Theo tính chất của dãy tỉ số bằng nha, ta có :
\(\dfrac{a_1}{a_2}=\dfrac{a_2}{a_3}=.....=\dfrac{a_n}{a_{n+1}}=\dfrac{a_1+a_2+....+a_n}{a_2+a_3+....+a_{n+1}}\)
\(\Rightarrow\dfrac{a_1}{a_2}=\dfrac{a_1+a_2+....+a_n}{a_2+a_3+....+a_{n+1}}\)
\(\dfrac{a_2}{a_3}=\dfrac{a_1+a_2+.....+a_n}{a_2+a_3+.....+a_{n+1}}\)
.................................
\(\dfrac{a_n}{a_{n+1}}=\dfrac{a_1+a_2+.....+a_n}{a_2+a_3+.....+a_{n+1}}\)
\(\Rightarrow\left(\dfrac{a_1+a_2+.....+a_n}{a_2+a_3+.....+a_{n+1}}\right)^n=\dfrac{a_1}{a_2}.\dfrac{a_2}{a_3}........\dfrac{a_n}{a_{n+1}}\)
Vậy \(\left(\dfrac{a_1+a_2+......+a_n}{a_2+a_3+......+a_{n+1}}\right)=\dfrac{a_1}{a_{n+1}}\) (đpcm)
~ Học tốt ~
\(a_1+a_2+a_3+..+a_{2015}=0\)\(0\)
\(\Rightarrow\left(a_1+a_2\right)+...+\left(a_1+a_{2015}\right)\)\(=\frac{\left(2015-1\right)}{2}+1=1008\)
\(\Rightarrow a_1+\left(a_1+a_2+..+a_{2015}\right)=1008\)
\(\Rightarrow a_1=1008\)
Ta có:
\(a_1+a_2+...+a_{2015}=0\)
\(\Leftrightarrow\left(a_1+a_2\right)+\left(a_3+a_4\right)+...+\left(a_{2013}+a_{2014}\right)+\left(a_{2015}+a_1\right)-a_1=0\)
\(\Leftrightarrow1+1+...+1-a_1=0\)
\(\Leftrightarrow1008-a_1=0\)
\(\Leftrightarrow a_1=1008\)
Ta có \(\frac{a_1}{a_2}=\frac{a_2}{a_3}=\frac{a_3}{a_4}=...=\frac{a_{2020}}{a_{2021}}=\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\)(dãy tỉ só bằng nhau)
=> \(\frac{a_1}{a_2}=\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\)
<=> \(\left(\frac{a_1}{a_2}\right)^{2020}=\left(\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\right)^{2020}\)
<=> \(\frac{a_1}{a_2}.\frac{a_1}{a_2}.\frac{a_1}{a_2}...\frac{a_1}{a_2}=\left(\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\right)^{2020}\)
<=> \(\frac{a_1}{a_2}.\frac{a_2}{a_3}.\frac{a_3}{a_4}...\frac{a_{2020}}{a_{2021}}=\left(\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\right)^{2020}\)
<=> \(\frac{a_1}{a_{2021}}=\left(\frac{a_1+a_2+a_3+...+a_{2020}}{a_2+a_3+a_4+...+a_{2021}}\right)^{2020}\)