Question

The largest term in the sequence \( a_n = \frac{n^2}{n^3 + 200} \) is given by

• A. $\frac{529}{49}$
• B. $\frac{8}{89}$
• C. $\frac{49}{543}$
• D. None of these

Hint:

For sequences, convert to function and check nearest integers after differentiation.

Answer 1

The Correct Option is C

To find the largest term in the sequence (\(a_n = \frac{n^2}{n^3 + 200}\)), we need to analyze and compare the value of the sequence for different values of \(n\).

1. Let's first understand the behavior of the function \(a_n\):
   • As \(n\) increases, the term \(n^3\) in the denominator grows much faster than the term \(n^2\) in the numerator, which means \(a_n\) will eventually decrease as \(n\) grows large.
2. To find the maximum term, we must consider small values of \(n\) where \(n^2\) and \(n^3 + 200\) are comparable.
3. We will check integer values of \(n\) starting from the smallest positive integer until a clear peak is identified:
   • For \(n = 1\), \(a_1 = \frac{1^2}{1^3 + 200} = \frac{1}{201}\).
   • For \(n = 2\), \(a_2 = \frac{2^2}{2^3 + 200} = \frac{4}{208} = \frac{1}{52}\).
   • For \(n = 3\), \(a_3 = \frac{3^2}{3^3 + 200} = \frac{9}{227}\).
   • For \(n = 4\), \(a_4 = \frac{4^2}{4^3 + 200} = \frac{16}{264} = \frac{1}{16.5}\).
   • For \(n = 5\), \(a_5 = \frac{5^2}{5^3 + 200} = \frac{25}{325} = \frac{1}{13}\).
   • For \(n = 6\), \(a_6 = \frac{6^2}{6^3 + 200} = \frac{36}{416} = \frac{9}{104}\).
   • For \(n = 7\), \(a_7 = \frac{7^2}{7^3 + 200} = \frac{49}{543}\).
4. Comparing the obtained values, \(a_7 = \frac{49}{543}\) is the largest value for the sequence \(a_n\) for \(n = 7\).

Thus, the answer is \(\frac{49}{543}\). Therefore, the correct option is \(\frac{49}{543}\).