To approximate \( \sqrt{1935} \) using a sequence of values that gets closer to \( 5 \), you can use the method of successive approximations. Here’s how to approach it:
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Understand that \( \sqrt{1935} \) is between two integers. Since \( 1935 \) is between \( 1936 \) and \( 1930 \), we know that \( \sqrt{1935} \) will be between \( 44 \) (since \( 44^2 = 1936 \)) and \( 43 \) (since \( 43^2 = 1849 \)).
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Start with an initial guess. A good starting point might be \( 44 \), because it is close to the upper bound.
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Use the average of your guess and the result of dividing the original number by your guess. This is known as the Babylonian method (also known as Heron's method).
- Let’s say your initial guess \( x_0 = 44 \).
- Calculate a new guess \( x_1 = \frac{x_0 + \frac{1935}{x_0}}{2} \).
- Substitute \( x_0 = 44 \):
- Calculate \( \frac{1935}{44} \approx 43.97727 \).
- Therefore, \( x_1 = \frac{44 + 43.97727}{2} \approx 43.98864 \).
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Repeat the process with \( x_1 \):
- Now, take \( x_1 = 43.98864 \) as your next guess.
- Calculate \( x_2 = \frac{x_1 + \frac{1935}{x_1}}{2} \).
- Substitute \( x_1 \):
- Calculate \( \frac{1935}{43.98864} \approx 43.99843 \).
- Therefore, \( x_2 = \frac{43.98864 + 43.99843}{2} \approx 43.99354 \).
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Continue this process until the guesses converge. Each iteration should get you closer to \( \sqrt{1935} \).
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For practical purposes, we can also quickly calculate:
- \( 44^2 = 1936 \)
- \( 43^2 = 1849 \)
Thus, we can narrow it down quickly:
- Approximating further:
- \( 43.99^2 \approx 1931.9601 \) (too low)
- \( 44.01^2 \approx 1934.4801 \) (too low)
- \( 44.05^2 \approx 1936.0025 \) (too high)
From this, we conclude that the approximation for \( \sqrt{1935} \) is about \( 44.0 \).
Final answer, correct to one decimal place: 44.0.
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