A seemingly simple but really interesting problem

**rickrishav** · November 3rd 2010, 07:28 PM

What will be the value of the following expression (May be in fractional form)?
$0.7 + 7.7 + 0.777 + 7.777 + 0.77777 + ………… t_2_0$

I tried to solve the problem in the following way:
0.7 + 7.7 + 0.777 + 7.777 + 0.77777 + ………… $t_2_0$
= 7/10 + 77/10 + 777/1000 + 7777/1000 ……
= [7/10 + 7 + 7/10] + [777/1000 + 7 + 777/1000]…..
= 7*10 + 2( 7/10 + 777/1000 + 77777/100000……. $t_1_0$ )
Is this approach correct and if so how should I proceed? If not how shall I attack the problem?

**Arka** · November 3rd 2010, 09:05 PM

$Hello Rick, I think it should be somewhat like this..... 7{(1/10) + (11/10) + (111/1000) + (1111/1000) +..........to 20 terms) =(7/9){(9/10) + (99/10) + (999/1000) +.........................to 20 terms) =(7/9)[{(10-1)/10} + {(100-1)/10} + {(1000-1)/1000}+...to 20 terms) =(7/9)[ 1-(1/10) + 10 - (1/10) +..................................to 20 terms) I think you can do it from here..............$

**Arka** · November 3rd 2010, 09:07 PM

Hello Rick,
I think it should be somewhat like this.....

$7{(1/10) + (11/10) + (111/1000) + (1111/1000) +..........to 20 terms) =(7/9){(9/10) + (99/10) + (999/1000) +.........................to 20 terms)$
$=(7/9)[{(10-1)/10} + {(100-1)/10} + {(1000-1)/1000}+...to 20 terms) =(7/9)[ 1-(1/10) + 10 - (1/10) +..................................to 20 terms)$
I think you can do it from here..............

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