Q.11. Evaluate

$$\sum_{n=1}^{100} \log_{10} \left( 3^{1-n} \right)$$

Answer Choices

• $-4950 \log_{10} 3$
• $4950 \log_{10} 3$
• $-5050 \log_{10} 3$
• $5050 \log_{10} 3$
• $1 - 4950 \log_{10} 3$
• $1 + 4950 \log_{10} 3$
• $1 - 5050 \log_{10} 3$
• $1 + 5050 \log_{10} 3$

Solution

We use the logarithmic property $\log_b(a^c) = c \log_b(a)$ to simplify the sum:

$$\log_{10} \left( 3^{1-n} \right) = (1 - n) \log_{10}(3)$$

This changes the sum to:

$$\sum_{n=1}^{100} (1 - n) \log_{10}(3) = \log_{10}(3) \sum_{n=1}^{100} (1 - n)$$

We split the sum:

$$\sum_{n=1}^{100} (1 - n) = \sum_{n=1}^{100} 1 - \sum_{n=1}^{100} n = 100 - 5050 = -4950$$

Thus, the sum becomes:

$$-4950 \log_{10}(3)$$

The correct answer is: $-4950 \log_{10}(3)$ .