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Evaluate: $1+(1+b)r+(1+b+b^2)r^2+\dots$ to infinite terms for $|br|< 1$.

  1. $\frac{1}{(1-br)(1-r)}$
  2. $\frac{1}{(1-r)(1-b)}$
  3. $\frac{1}{(1-b)(b-r)}$
  4. None of these
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$1 + (1 + b)r + (1 + b + b^2)r^2 + (1 + b + b^2 + b^3)r^3 + ... \infty \\ = \sum_{i=0}^{\infty} (\sum_{j=0}^{i} b^j)r^i\\ = \sum_{i=0}^{\infty} (\frac{1-b^{i+1}}{1-b})r^i\\ =\frac{1}{1-b} \sum_{i=0}^{\infty} (r^i - b^{i+1}r^i) \\ = \frac{1}{1-b} ( \sum_{i=0}^{\infty} r^i - b\sum_{i=0}^{\infty} b^i r^i)\\ =\frac{1}{1-b} \times (\frac{1}{1-r} - \frac{b}{1-br})\\ = \frac{1}{1-b} \times \frac{1-b}{(1-r)(1-br)} \\ =\frac{1}{(1-r)(1-br)}$
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