Resistors in infinite series

For a > 1 : R = τ ( a ) = n = 0 1 a n = a a 1 a n d V = I R \text{For} \space a > 1: \\ R = \tau (a) = \sum_{n=0}^{\infty} \frac{1}{a^n} = \frac{a}{a-1} \\ and \\ V = IR

A circuit consisting of an infinite series of resistors is prepared such that the simplified equivalent resistor has a resistance ( R R ) equal to τ ( a ) \tau (a) . Determine the voltage ( V \text{V} ) a circuit that has a resistance of τ ( 5 ) Ω \tau (5) \space Ω and a current ( I \text{I} ) of 16 A 16 \space A .

Give your answer in volts.


The answer is 20.

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1 solution

David Hontz
Jun 14, 2016

R = τ ( 5 ) = n = 0 1 5 n = 5 4 V = I R = 16 × 5 4 = 80 4 = 20 v o l t s R = \tau(5) = \sum_{n=0}^{\infty} \frac{1}{5^n} = \frac{5}{4} \\ V = IR = 16 \times \frac{5}{4} = \frac{80}{4} = \boxed{20 \space volts}

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