Question

Find the value of \(\log_{10}10+\log_{10}10^2+\log_{10}10^3+\ldots+\log_{10}10^n\)

• A. \(n^2+1\)
• B. \(n^2-1\)
• C. \(\dfrac{n(n+1)}{3}\)
• D. \(\dfrac{n^2+n}{2}\)

Hint:

Each log₁₀(10^k) simplifies to just k, turning the sum into 1+2+…+n.

Answer 1

The Correct Option is D

Recognising the log simplification early turns this into a very familiar sum.

1. Since $\log_{10}(10^k) = k$ for any power $k$, the whole expression collapses to $1+2+3+\dots+n$.
2. This is the standard sum of the first $n$ natural numbers, with a well-known closed form $\frac{n(n+1)}{2}$.
3. Expanding, this equals $\frac{n^2+n}{2}$, matching option D exactly.

So the correct answer is option D.