SOLUTION: The following terms √3x + √9x + √27x + ... form a pattern that continues until the sixth is found. 1. Compute the sum of all six terms...which I have done. 2.

Algebra ->  Square-cubic-other-roots -> SOLUTION: The following terms √3x + √9x + √27x + ... form a pattern that continues until the sixth is found. 1. Compute the sum of all six terms...which I have done. 2.       Log On


   

Question 706048: The following terms √3x + √9x + √27x + ... form a pattern that continues until the sixth is found.
1. Compute the sum of all six terms...which I have done.
2. By investigating the pattern further, develop a system that will enable you to find the sum of 12 such terms without actually writing them. I have written out the twelve terms and I cannot figure out a system. I would love a hint or clue to help me solve this.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


This is the most interesting problem I've seen on this site in a long time.

The first thing to do is to factor out of all of the terms, so that just becomes a factor in the answer.

Then you have a sum of radicals. Note that each of the radicands is one power of 3 larger than the previous. The pattern becomes evident when you replace the radical with a fractional exponent.







and so on...

So you could write the first six terms thus:



Now if you rearrange the terms so that the rational terms are in one group and the irrational terms are in another group:



Then factor the radical out of the first grouping (putting the back in):



The sum becomes trivial

So for 12 terms:



But then we get to make it even simpler because



from which we can derive



So your 12 term sum becomes



John

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