A geometric sequence is a sequence of numbers where each successive number is the product of the previous number and some constant r, i.e. an+1=rana_{n+1}=ra_n.  The constant factor rr is called the common ratio. For example, the following is a geometric sequence: {12,−16,118,−154,1162,⋯}\left\lbrace \frac{1}{2}, -\frac{1}{6}, \frac{1}{18}, -\frac{1}{54}, \frac{1}{162}, \cdots\right \rbrace. Thus, the common ratio r=r=[Fill in the blank], . Please fill in your answer as a fraction with no space. A geometric series is the sum of the terms of a geometric sequence. Thus, 12−16+118−154+1162−⋯=∑n=0∞arn\frac{1}{2}-\frac{1}{6}+\frac{1}{18}-\frac{1}{54}+\frac{1}{162}-\cdots= \displaystyle \sum_{n=0}^{\infty} ar^n . Here, a=a=[Fill in the blank], . Please fill in your answer as a fraction with no space. Find ∑n=0∞rn= \displaystyle \sum_{n=0}^{\infty} r^n =[Fill in the blank], . Please fill in your answer as a fraction with no space. Therefore, 12−16+118−154+1162−⋯=∑n=0∞arn=a∑n=0∞rn=\frac{1}{2}-\frac{1}{6}+\frac{1}{18}-\frac{1}{54}+\frac{1}{162}-\cdots= \displaystyle \sum_{n=0}^{\infty} ar^n =a \displaystyle \sum_{n=0}^{\infty} r^n = [Fill in the blank], . Please fill in your answer as a fraction with no space. Multiple fill-in-the-blank