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If n is a positive integer such that \((10−−√7)(10−−√7)2…(10−−√7)n > 999\), then the smallest value of n is

Updated On: Jan 15, 2026
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Correct Answer: 6

Solution and Explanation

The provided inequality is:

$$(\sqrt{10} - \sqrt{7})(\sqrt{10} - \sqrt{7})^2 \cdots (\sqrt{10} - \sqrt{7})^n > 999$$

This simplifies to:

$$(\sqrt{10} - \sqrt{7})^{1 + 2 + \cdots + n} > 999$$

The exponent is the sum of the first \( n \) natural numbers, which is:

$$\frac{n(n+1)}{2}$$

Let \( x = \sqrt{10} - \sqrt{7} \). Approximating the values:

  • \( \sqrt{10} \approx 3.162 \)
  • \( \sqrt{7} \approx 2.646 \)
  • Therefore, \( x \approx 0.516 \)

The inequality now is:

$$x^{\frac{n(n+1)}{2}} > 999$$

Taking the natural logarithm of both sides:

$$\frac{n(n+1)}{2} \cdot \ln(0.516) > \ln(999)$$

Approximating the logarithms:

  • \( \ln(0.516) \approx -0.6618 \)
  • \( \ln(999) \approx 6.907 \)

Solving for the exponent:

$$\frac{n(n+1)}{2} > \frac{6.907}{-0.6618}$$

$$\frac{n(n+1)}{2} < -10.44$$

This result is a contradiction, as the left side of the inequality is positive, while the right side is negative.

Therefore, we test values of \( n \) to find the smallest integer that satisfies the original inequality.

  • Using \( x = 0.516 \)
  • For \( n = 6 \), the exponent is \( \frac{6 \cdot 7}{2} = 21 \)
  • \( x^{21} = (0.516)^{21} \approx 1010.2 \)
  • Since \( 1010.2 > 999 \), \( n=6 \) is a valid solution.

The smallest integer value for \( n \) is: \( \boxed{6} \)

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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