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March 4th, 2016, 12:12 PM   #1
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Geometric sequence

This problem keeps bothering me:
The sum of the first three terms of a geometric sequence is 23 3/4 (23.75) and the sum of the first four terms is 40 5/8. Find the first term and the common ratio.

So I found that the fourth term is 16 7/8 (16.875). And then the sum of the first three terms of this GP would be:
16.875/r^3+16.875/r^2+16.875/r=23.75
I know that the common ratio is 1.5. But I am stuck here; how should I solve this equation of the third degree?

And I do not think that I should approach this problem in this particular way. Is there any other way that makes more sense?
Thank you.

Last edited by skipjack; March 4th, 2016 at 12:33 PM.
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March 4th, 2016, 01:20 PM   #2
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I think your way is best.

Multiplying your equation by 8r³/5 gives 27 + 27r + 27r² = 38r³,
so 38r³ - 27r² - 27r - 27 = 0, i.e., (2r - 3)(19r² + 15r + 9) = 0.

Hence r = 3/2 or -3(5 ± $\small\sqrt{51}i$)/38.
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March 4th, 2016, 11:30 PM   #3
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Thank you for the response. Can you explain me this part?
38r³ - 27r² - 27r - 27 = 0, i.e., (2r - 3)(19r² + 15r + 9) = 0.
How we get the second part?

Last edited by skipjack; March 5th, 2016 at 03:55 AM.
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March 5th, 2016, 03:53 AM   #4
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The rational root can be found with the help of the rational root theorem.
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