prove: 1 + 11 + 111 + .... + ((111 ...111)/(n times)) = ((10^(n + 1) − 9n − 10)/(81))


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Question Number 44716 by Tawa1 last updated on 03/Oct/18

$prove : 1 + 11 + 111 + . . . . + \frac{111 . . . 111}{n times} = \frac{10^{n + 1} - 9 n - 10}{81}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$s = 1 + 11 + 111 + 1111 + . . . + 111 . . \underset{n t i m e s}{.} 111$ $9 s = 9 + 99 + 999 + 9999 + . . . + 999 . . . 999$ $9 s = (10 - 1) + (10^{2} - 1) + (10^{3} - 1) + (10^{4} - 1) + . . + (10^{n} - 1)$ $9 s = (10 + 10^{2} + 10^{3} + . . + 10^{n}) - (1 + 1 + 1 . . . n t i m e s)$ $9 s = \frac{10 (10^{n} - 1)}{10 - 1} - n$ $s = \frac{10^{n + 1} - 10 - 9 n}{81}$ $p r o o f b y m a t h e m a t i c a l i n d u c t i o n$ $s t e p 1 p u t n = 1 r i g h t h a n d s i d e$ $o b t a i n e d v a l u e i n R H S i s \frac{10^{2} - 10 - 9 \times 1}{81} = 1$ $n o w i n L H S v a l u e a l s o 1$ $s o g i v e n s t a t e m e n t i s t r u e f o r n = 1$ $s t e p 2 p u t n = 2$ $R H S = \frac{10^{2 + 1} - 10 - 9 \times 2}{81} = \frac{1000 - 10 - 18}{81} = 12$ $L H S i s a l s o 1 + 11 = 12$ $s o f o r n = 2 i t i s t r u e$ $s t e p 3$ $l e t w e a s s u m e f o r n = m i t i s t r u e$ $t h e n w e h a v e t o p r o v e t h a t i t i s t r u e f o r m + 1$ $1 + 11 + 111 + . . . + 111 . . 111 \leftarrow m t i m e s = \frac{10^{m + 1} - 10 - 9 m}{81}$ $n o w a d d 111 . . . 111 \leftarrow (m + 1) t k m e s i n L H S$ $\frac{10^{m + 1} - 10 - 9 m}{81} + 111 . . . 111 \leftarrow (m + 1) t i m e s$ $= \frac{10^{m + 1} - 10 - 9 m}{81} + \frac{1}{9} (999 . . . 999) \leftarrow (m + 1) t i m e s$ $= \frac{10^{m + 1} - 10 - 9 m}{81} + \frac{1}{9} (10^{m + 1} - 1)$ $= \frac{10^{m + 1} - 10 - 9 m + 9 (10^{m + 1} - 1)}{81}$ $= \frac{10^{m + 1} - 10 - 9 m + (10 - 1) (10^{m + 1} - 1)}{81}$ $= \frac{10^{m + 1} - 10 - 9 m + 10^{m + 2} - 10 - 10^{m + 1} + 1}{81}$ $= \frac{10^{m + 2} - 10 - 9 m - 10 + 1}{81}$ $= \frac{10^{m + 2} - 10 - 9 m - 9}{81}$ $= \frac{10^{m + 2} - 10 - 9 (m + 1)}{81}$ $i f y o u p u t n = m + 1 R H S$ $\frac{10^{m + 2} - 10 - 9 (m + 1)}{81} s o g i v e n s t a t e m e n t a s o t r u e$ $f o r m + 1$ $s o i t t r u e f o r m q n d m + 1 . . .$
	Commented by Tawa1 last updated on 03/Oct/18

	$God bless you sir .$
	Commented by Tawa1 last updated on 03/Oct/18

	$Please sir, can you help me to use mathematical induction sir$
	Commented by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$y e s . . .$
	Commented by Tawa1 last updated on 03/Oct/18

	$God bless you sir . help me to use induction$
	Commented by Tawa1 last updated on 04/Oct/18

	$God bless you sir . I really appreciate your effort .$
	Commented by Necxx last updated on 04/Oct/18

	$w o w . . . . I^{'} v e a l s o s o u g h t t h e a n s w e r$ $f o r t h i s . T h a n k s M r T a n m a y .$
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