Algebra #26
by djmathman, Oct 29, 2011, 12:35 AM
Problem wrote:
Solve 
for all real values of 
.
(SOURCE: NYSML)
for all real values of 
.(SOURCE: NYSML)
There isn't much we can do at the moment, so we try to simplify the expression. Using our log rules, we can rewrite the equation as
. Since our barrier at the moment is that pesky 
, we focus only on that summation and leave the 
out for a moment. The first couple of terms of this summation will be 
, 
, 
. This is something we are familiar with! Using the formula for a geometric series and simplifying (or just memorizing the form for the sum of the first powers of 
starting from 
), we can write the sum as 
. Since 
goes up to 
, the sum is thus 
.Now we can rid the summation from the equation to get
. To solve for , we first subtract from both sides to get 
. Dividing by to isolate , we get 
. At first the answer seems to be complicated, but luck turns its tide when we realize that 
. Thus, we can simplify the LHS to 
. Finally, we have 
, and solving for gives us 
.Overall I thought this was a really interesting problem.
This post has been edited 4 times. Last edited by djmathman, Dec 2, 2011, 11:11 PM
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