Discussion
If \(\sum_{k=1}^{n} \log_{10}(5^{k}) = 66\log_{10}(5)\), then the value of n is equal to
Answer: D
n = 11.
Use \(\log(5^{k}) = k\log 5\):
\[ \begin{aligned} \sum_{k=1}^{n} k\log_{10}5 &= \frac{n(n + 1)}{2}\log_{10}5 \\ \frac{n(n + 1)}{2} &= 66 \\ n(n + 1) &= 132 = 11 \times 12 \end{aligned} \]
So n = 11. Check the neighbours: n = 10 gives 55 and n = 12 gives 78, so only 11 makes the sum 66 times \(\log 5\).
The key step is the log rule \(\log(a^{k}) = k\log a\), which turns the sum of logarithms into \(\log 5\) times the sum 1 + 2 + ... + n.
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