` A = 1 - 3/4 + (3/4)^2 - (3/4)^3 + (3/4)^4 - ... - (3/4)^{2009} + (3/4)^{2010} `
` => 3/4 A = 3/4 - (3/4)^2 + (3/4)^3 - (3/4)^4 + (3/4)^5 - ... - (3/4)^{2010} + (3/4)^{2011} `
` => A + 3/4 A = 1 - 3/4 + (3/4)^2 - (3/4)^3 + (3/4)^4 - ... - (3/4)^{2009} + (3/4)^{2010} `
` => 7/4 A = 1 + 3/4 - 3/4 + (3/4)^2 - (3/4)^2 + ... + (3/4)^{2010} - (3/4)^{2010} + (3/4)^{2011} `
` => 7/4 A = 1 + (3/4)^{2011} `
` => A = 4/7 + 4/7 . 3/4 . (3/4)^{2010} `
` => A = 4/7 + 3/7 . (3/4)^{2010} `