Math Problem Solver

\[ \frac{2}{9} + \frac{2}{126} + \frac{2}{168} + \frac{2}{216} + \frac{2}{270} + \frac{2}{330} = \frac{3}{11} = 0.\overline{27} \]

The sum can be computed by rewriting each term over a common denominator, where the least common denominator of \(\frac{2}{9}\), \(\frac{2}{126}\), \(\frac{2}{168}\), \(\frac{2}{216}\), \(\frac{2}{270}\), and \(\frac{2}{330}\) is \(83160\). \[ \begin{aligned} \frac{2}{9} + \frac{2}{126} + \frac{2}{168} + \frac{2}{216} + \frac{2}{270} + \frac{2}{330} &= \frac{2 \cdot 9240}{9 \cdot 9240} + \frac{2 \cdot 660}{126 \cdot 660} + \frac{2 \cdot 495}{168 \cdot 495} + \frac{2 \cdot 385}{216 \cdot 385} + \frac{2 \cdot 308}{270 \cdot 308} + \frac{2 \cdot 252}{330 \cdot 252} \\ &= \frac{18480 + 1320 + 990 + 770 + 616 + 504}{83160} \\ &= \frac{22680}{83160} \\&= \frac{3}{11} \end{aligned} \]