Find the sum of the series upto n terms `0.b+0.b b+0.b b b +0.b b b b+"......",AA " b" in N " and "1 le b le 9.`

Explanation

Let `S=0.b+0.b b+0.b b b +0.b b b b+"......""upto n terms"`
`=b(0.1+0.11+0.111 +0.1111+"......""upto n terms")`
`=(b)/(9)(0.9+0.99+0.999+0.9999+"......""upto n terms")`
`=(b)/(9){(1-0.1)+(1-0.01)+(1-0.001)+(1-0.0001)+"......""upto n terms")}`
`=(b)/(9){(1+1+1+1)+"......""upto n terms")-(0.1+0.01+0.001+0.0001+"......""upto n terms")}`
`=(b)/(9){n-((1)/(10)+(1)/(10^(2))+(1)/(10^(3))+(1)/(10^(4))+"......""upto n terms")}`
`=(b)/(9){n-(((1)/(10)(1-((1)/(10))^(n)))/(1-(1)/(10))}=(b)/(9){n-(1)/(9)[1-((1)/(10))^(n)]}`