SOLUTION: Find the sum of the following sequence : 2, 22, 222, ......... to n terms.

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Question 1088866: Find the sum of the following sequence :
2, 22, 222, ......... to n terms.
Answer by ikleyn(52781) About Me  (Show Source):
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Let S%5Bn%5D = 2 + 22 + 222 + . . . + the last n-th term.


Then %289%2F2%29%2AS%5Bn%5D = 9 + 99 + 999 + . . . + the last term written with n "9".


The k-th term plus 1 is equal to 10%5Ek.  Therefore,

%289%2F2%29%2AS%5Bn%5D = 10+%2B+10%5E2+%2B+10%5E3+%2B+.+.+.+%2B+10%5En - n.


Next, apply the formula for the sum of geometric progression 10+%2B+10%5E2+%2B+10%5E3+%2B+ellipsis+%2B+10%5En.

On geometric progressions,  see the introductory lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Geometric progressions".