SOLUTION: sum of infinite gp is 2 and sum of their cubes of the terms is 8/7 find gp

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Question 1112554: sum of infinite gp is 2 and sum of their cubes of the terms is 8/7 find gp
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
The first equation needed is:
+sum%28a%2Ar%5En%2C+0%2C+infinity%29+=++a%2F%281-r%29+ = 2
The 2nd equation may not be as obvious but if you substitute +A=a%5E3+ and R=r%5E3 into the infinite sum formula for a GP, you will find:
+sum%28a%5E3%2Ar%5E%283n%29%2C+0%2C+infinity%29+=+a%5E3%2F%281-r%5E3%29+ = 8/7

These two equations can be solved for
+highlight%28a=1%29+ and +highlight%28+r=1%2F2+%29+
———————————
The work-out:
+a%2F%281-r%29+=+2+ —> +a+=+2-2r+
+a%5E3%2F%281-r%5E3%29+=+8%2F7+ —> +a%5E3+=+%288%2F7%29%281-r%5E3%29+
So +%282-2r%29%5E3++=+%288%2F7%29%281-r%5E3%29+
…. after some algebra ...
++-6r%5E3+%2B21r%5E2+-+21r+%2B+6+=+0+
This has solutions +r=1%2F2+, +r=1+ and +r=2+
but only +r=1%2F2+ maintains +abs%28r%29+%3C+1+ (as required for convergence of infinite GP)
+a+=+2-2r+=+2-2%281%2F2%29+=+1+