Find the sum upto n terms of the series `a+aa+aaa+aaaa+"......",AA a in N " and "1 le a le 9.`

Explanation

Let ` S = a+aa+aaa+aaaa+"......" " upto n terms " `
` = a(1+11+111+1111+"......" "upto n terms") `
` = (a)/(9)(9+99+999+9999+"......" "upto n terms") `
` = (a)/(9){(10^(1)-1)+(10^(2)-1)+(10^(3)-1)+(10^(4)-1)+"......" "upto n terms"} `
` = (a)/(9){10+10^(2)+10^(3)+"......" "upto n terms")-(1+1+1+"......" "upto n terms")}`
` = (a)/(9){(10(10^(n)-1))/(10-1)-n}=(a)/(9){(10)/(9)(10^(n)-1)-n}`