$\\$
Đặt `a_1/a_2 = a_2/a_3=...=a_{2008}/a_{2009}=k (k \ne 0)`
`->a_1=a_2k, a_2=a_3k,..., a_{2008}=a_{2009}k`
`a_1/a_{2009}`
`= (a_1 . a_2 ... a_{2008})/(a_2 . a_3 ... a_{2009})`
`= (a_2k . a_3k ... a_{2009}k)/(a_2.a_3...a_{2009})`
`= ((a^2.a_3...a_{2009}) (k.k...k) )/(a_2 . a_3 ... a_{2009})`
`= k^{2008}` (1)
`( (a_1+a_2+...+a_{2008})/(a_2+a_3+...+a_{2009}) )^{2008}`
`= ( (a_2k +a_3k + ... +a_{2009}k)/(a_2+a_3+...+a_{2009}) )^{2008}`
`= ( (k (a_2+a_3+...+a_{2009}) )/(a_2+a_3+...+a_{2009}) )^{2008}`
`= k^{2008}` (2)
Từ (1), (2)
`-> a_1/a_{2009}=( (a_1+a_2+...+a_{2008})/(a_2+a_3+...+a_{2009}) )^{2008} (=k^{2008})`
`->` Đpcm
