Ta có: `(a_1)/(a_2) = (a_2)/(a_3) =....= (a_2010)/(a_2011)`
`=> (a_1)/(a_2). (a_2)/(a_3) ...(a_2010)/(a_2011)`
`= (a_1)/(a_2011) (1)`
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
`(a_1)/(a_2) = (a_2)/(a_3) =....= (a_2010)/(a_2011)`
`= (a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011)`
`=>(a_1)/(a_2) = (a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011)`
`(a_2)/(a_3) = (a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011)`
`.................................................................`
`(a_2010/a_2011)= (a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011)`
Do đó : `(a_1)/(a_2). (a_2)/(a_3) ...(a_2010)/(a_2011) = ((a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011))^2010 (2)`
Từ `(1); (2)` `=> (a_1)/(a_2011) = ((a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011))^2010 `
Vậy `(a_1)/(a_2011) = ((a_1 + a_2 + a_3 +...+a_2010)/(a_2 + a_3 +...+a_2011))^2010 `
