Question

find the sum of series 0.7+0.77+0.777+.... upto n terms​

Answer 1

Answer:

$s = 0.7 + 0.77 + 0.777 + ...... \\ \\ = 7(0.1 + 0.11 + 0.111 + ......) \\ \\ = \frac{7}{9} \times 9(0.1 + 0.11 + 0.111 + .....) \\ \\ = \frac{7}{9} (0.9 + 0.99 + 0.999 + ....) \\ \\ = \frac{7}{9} (10 - 0.1 + 100 - 0.01 + 1000 - 0.001...) \\ \\ = \frac{7}{9} ((10 + 100 + 1000...) - ( \frac{1}{10} + \frac{1}{100} + ...)$

Use formula of sum of n terms of GP

$s = \frac{a( {r}^{n} - 1) }{r - 1}$

$\frac{7}{9} ( \frac{10( {10}^{n} - 1) }{10 - 1} - \frac{ \frac{1}{10}( {10}^{ - n} - 1) }{ \frac{1}{10} - 1 } ) \\ \\ = \frac{7}{9} ( \frac{10( {10}^{n} - 1) }{9} - \frac{ 1 - {10}^{ - n} }{9} ) \\ \\ = \frac{7}{81} ( {10}^{n + 1} + {10}^{ - n} - 11)$