If 5+55+555+⋯to n terms=581(10n+1+x+yn), then the value of y−x is
A
1
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B
−19
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C
−1
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D
19
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Solution
The correct option is A1 Let Sn denote the sum to n terms of the given series, then Sn=5+55+555+⋯to n terms⇒Sn=5[1+11+111+⋯to n terms]⇒Sn=59[9+99+999+⋯to n terms]⇒Sn=59[(10−1)+(102−1)+(103−1)+⋯(10n−1)]⇒Sn=59[(10+102+103+⋯10n)−n]⇒Sn=59[10×(10n−1)10−1−n]⇒Sn=59[109×(10n−1)−n]⇒Sn=581[10n+1−10−9n]⇒x=−10,y=−9∴y−x=1