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16 Sep 2014, 01:05
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Is \(x + x^2 + ... + x^9 + x^{10}\) positive?
(1) \(x < -1\)
(2) \(x^2 > 2\)
(1) \(x < -1\)
(2) \(x^2 > 2\)
16 Sep 2014, 01:05
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Official Solution:
Is \(x+x^2+ ...+x^9+x^{10} \gt 0\)?
(1) \(x < -1\).
From this statement, we can deduce that each pair of terms, such as \(x+x^2\), \(x^3+x^4\), and so on up to \(x^9+x^{10}\), is positive. Consequently, their combined sum is positive. Sufficient.
(2) \(x^2 > 2\).
From this statement, we can deduce that \(x < -\sqrt{2}\) or \(x > \sqrt{2}\). If \(x > \sqrt{2}\), then obviously the sum is positive. If \(x\) is negative, then as noted above, each pair of terms, such as \(x+x^2\), \(x^3+x^4\), and so on up to \(x^9+x^{10}\), is positive. Consequently, their combined sum is positive. Sufficient.
Answer: D
Is \(x+x^2+ ...+x^9+x^{10} \gt 0\)?
(1) \(x < -1\).
From this statement, we can deduce that each pair of terms, such as \(x+x^2\), \(x^3+x^4\), and so on up to \(x^9+x^{10}\), is positive. Consequently, their combined sum is positive. Sufficient.
(2) \(x^2 > 2\).
From this statement, we can deduce that \(x < -\sqrt{2}\) or \(x > \sqrt{2}\). If \(x > \sqrt{2}\), then obviously the sum is positive. If \(x\) is negative, then as noted above, each pair of terms, such as \(x+x^2\), \(x^3+x^4\), and so on up to \(x^9+x^{10}\), is positive. Consequently, their combined sum is positive. Sufficient.
Answer: D
23 Feb 2021, 11:44
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If 1<x<0, then the equation won't be positive, correct?
