Hello, thereddevils!
This is not an easy problem . . .
The general term of this sequence is tricky to derive.Find the term and the sum of the first terms of the series:
. . . . . .
The 5th term has five 1's.
How can we "create" the number 11,111 from a formula?
We can get five 9's from: . . . . then divide by 9.
. . That is: .
Therefore, the term is: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The sum of the first terms is: .
. . which equals: .
The first sum is: .
. . . This is a geometric series whose sum is: .
The second sum is: .
So we have: .
. . Therefore: .