Evaluate :
(i) ` 1 + 2 + 4 + …` to 10 terms " " `(ii) sqrt(2) - 2 + 2 sqrt(2) - …+ 64 sqrt(2)`

Explanation

(i) The series is ` 1 + 2 + 4 + …` to 10 terms
Here , `(a_(2))/(a_(1)) = (a_(3))/(a_(2)) = …= 2 `
Thus, given series is a geometrix series with a = 1 and r = 2
` therefore ` Required sum ` S_(10)(1(2^(10) - 1))/(2-1) = (1024 - 1)/(1) = 1023`
(ii) The series is ` sqrt(2) - 2 + 2 sqrt(2) - ... + 64 sqrt(2)`
Here , `(a_(2))/(a_(1)) = (a_(3))/(a_(2)) = …= -sqrt(2 )`
Thus, given series is a geomeric series with ` a = sqrt(2), r = - sqrt(2)` and last term `I = 64sqrt(2)`
`therefore` Requred sum `(sqrt(2)-64sqrt(2)(-sqrt(2)))/(1-(-sqrt(2)))=(sqrt(2)+128)/(sqrt(2)+1)xx(sqrt(2)-1)/(sqrt(2)-1)`
`(2-sqrt(2)+128 sqrt(2) - 128)/(2-1) = 127sqrt(2) - 126`