Question 1088866
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<pre>
Let {{{S[n]}}} = 2 + 22 + 222 + . . . + the last n-th term.


Then {{{(9/2)*S[n]}}} = 9 + 99 + 999 + . . . + the last term written with n "9".


The k-th term plus 1 is equal to {{{10^k}}}.  Therefore,

{{{(9/2)*S[n]}}} = {{{10 + 10^2 + 10^3 + . . . + 10^n}}} - {{{n}}}.


Next, apply the formula for the sum of geometric progression {{{10 + 10^2 + 10^3 + ellipsis + 10^n}}}.
</pre>

On geometric progressions, &nbsp;see the introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-geometric-progressions.lesson>Problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-geometric-progressions.lesson>Word problems on geometric progressions</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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