Step 1: Understanding the given sum The given sum in the numerator is: \[ S = 17^7 + 27^7 + \dots + n^{77}. \] The number of terms in the sum is approximately \( n \), and the dominant term in the summation is: \[ \sum_{k=1}^{n} k^{77}. \] Using the standard asymptotic sum formula: \[ \sum_{k=1}^{n} k^m \approx \frac{n^{m+1}}{m+1}, \] for large \( n \), we approximate: \[ \sum_{k=1}^{n} k^{77} \approx \frac{n^{78}}{78}. \] Step 2: Evaluating the limit Substituting the approximation: \[ \lim_{n \to \infty} \frac{\sum_{k=1}^{n} k^{77}}{n^{78}} = \lim_{n \to \infty} \frac{\frac{n^{78}}{78}}{n^{78}}. \] \[ = \frac{1}{78}. \] Step 3: Conclusion Thus, the correct answer is: \[ \frac{1}{78}. \]
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Approach Solution -2
To evaluate the limit \( \lim_{n \to \infty} \frac{17^7 + 27^7 + \dots + n^{77}}{n^{78}} \), we start by analyzing the series in the numerator. Consider the general term \( k^{77} \) where \( k \) ranges from 17 to \( n \). The numerator can be rewritten as a sum:
\( \sum_{k=17}^{n} k^{77} \).
The key is to approximate this sum using integral calculus for large \( n \). The sum \( \sum_{k=17}^{n} k^{77} \) can be closely approximated by the integral:
