Statement -1 For all natural numbers n , `0.5+0.55+0.555+......` upto n terms `=(5)/(9){n-(1)/(9)(1-(1)/(10^n))}` Statement-2 `a+ar+ar^2+....+ar^(n-1)=(a(1-r^n))/((1-r))`, for `0lt r lt 1` .

Options

A. Statement -1 is true , Statement -2 is true, Statement -2 is correct explanation for Statement -1
B. Statement -1 is true , Statement -2 is true , Statement -2 is not correct explanation for Statement -1
C. Statement -1 is true , Statement -2 is false
D. Statement -1 is false , Statement - 2 is true.

Explanation

Step I For n=1.
LHS `=0.5` and RHS `=(5)/(9){1-(1)/(9)(1-(1)/(10))}=(5)/(9)(1-(1)/(10))=(5)/(10)=0.5`
`therefore LHS=RHS`
which is true for `n=1`.
Step II Assume it is true for `n=k` , then `0.5+0.55+0.555+.....+` upto k terms
`=(5)/(9){k-(1)/(9)(1-(1)/(10^k))}`
Step III For `n=k+1`,
LHS `=0.5+0.55+0.555+....+` upto `(k+1)` terms
`=(5)/(9){k-(1)/(9)(1-(1)/(10^k))}+(k+1)` th terms
`=(5)/(9){k-(1)/(9)(1-(1)/(10^k))}+ubrace(0.555....5)_((k+1)"terms")`
`=(5)/(9){k-(1)/(9)(1-(1)/(10^k))}+(1)/(10^(k+1))(ubrace(555....5)_((k+1)"terms"))`
`=(5)/(9){k-(1)/(9) (1-(1)/(10^k))}+(5)/(10^(k+1))(1+10+10^2+.....+10^k)`
`=(5)/(9) {k-(1)/(9)(1-(1)/(10^k))}+(5.(10^(k+1)-1))/(10^(k+1).(10-1))`
`=(5)/(9){k-1(1)/(9)(1-(1)/(10^k))+(10^(k+1)-1)/(10^(k+1))}`
`=(5)/(9){(k+1)-(1)/(9) +(1)/(9.10^k)-(1)/(10^(k+1))}`
`=(5)/(9){(k+1)-(1)/(9) +((10-9))/(9.10^(k+1))}`
`=(5)/(9){(k+1)-(1)/(9)(1-(1)/(10^(k+1))}=RHS`
which is true for `n=k+1`.
Hence , bothe statements are true but Statement- 2 is not a correct explanation for Statement-1.