prove that 5 +55 +555 +5555..... n digits =5 /81 (10 per power n+1_9n_

1.	prove that 5 +55 +555 +5555..... n digits =5 /81 (10 per power n+1_9n_10)
Answer» Let Sn = 5 + 55 + 555 + ........... to n terms then,Sn = 5 ( 1 + 11 + 111 +........ to n terms )Sn = 5/9 ( 9 + 99+999+.......to n terms )Sn = 5/9 { ( 10-1 ) + ( 100-1 ) + ( 1000-1 )+.......+ ( 10 to the power n -1 )}Sn = 5/9 { ( 10+100+1000 +........ 10 to the power n ) }Sn = 5/9 { 10 (10 to the power n - 1/ 10-1) - n}Sn = 5/9{ 10/9 ( 10 to the power n. - 1 ) - n} Sn = 5/81 ( 10 to the power n+1 -9n -10 )

Discussion