Sum with natural log

Question

The sequence $u_n$ is defined by $u_n=0.5^n$

1. Find the exact value of $\sum_{0}^{10} ln(u_r)$

Is there a common ratio for this?

$\frac{ln(0.5^1)}{ln(0.5^0)}$ dividing by 0 not possible..?

Answer 1

You want $$\begin{align} \sum_{r=0}^{10} \ln(0.5^r) &= \ln\left[\left(\frac{1}{2}\right)^0\left(\frac{1}{2}\right)^1 \dots \left(\frac{1}{2}\right)^{10}\right] \\ &= \ln\left(\frac{1}{2^0\cdot 2^1 \cdots 2^{10}}\right)\\ &= -\ln(2^0\cdot2^1 \cdots 2^{10})\\ &= -\ln(2^{0 +1 + \dots + 10})\\ &= \dots \end{align} $$ We get this by using that $\ln(a) + \ln(b) = \ln(ab)$. All you have to find now is $1 + \dots + 10$.