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.

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) (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

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

The Out Campaign: Scarlet Letter of Atheism

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SOLUTION: The following terms &#8730;3x + &#8730;9x + &#8730;27x + ... form a pattern that continues until the sixth is found. 1. Compute the sum of all six terms...which I have done. 2.

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.