7. (a) Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ (b) Evaluate \(\quad \sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)\).