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18 November, 09:14

How many different messages can be transmitted in n microseconds using three different signals if one signal requires 1 microsecond for transmittal, the other two signals require 2 microseconds each for transmittal, and a signal in a message is followed immediately by the next signal?

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  1. 18 November, 11:17
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    a_{n} = 2/3. 2^n + 1/3. (-1) ^n

    Explanation:

    Let a_{n} represents number of the message that can transmitted in n microsecond using three of different signals.

    One signal requires one microsecond for transmittal: a_{n}-1

    Another signal requires two microseconds for transmittal: a_{n}-2

    The last signal requires two microseconds for transmittal: a_{n}-2

    a_{n} = a_{n-1} + a_{n-2} + a_{n-2} = a_{n-1} + 2a_{n-2}, n ≥ 2

    In 0 microseconds. exactly 1 message can be sent: the empty message.

    a_{0} = 1

    In 1 microsecond. exactly 1 message can be sent (using the one signal of one microseconds:

    a_{0} = 1

    2 - Roots Characteristic equation

    Let a_{n} = r^2, a_{n-1}=r and a_{n-2} = 1

    r^2 = r+2

    r^2 - r - 2 = 0 Subtract r+6 from each side

    (r - 2) (n+1) = 0 Factorize

    r - 2 = 0 or r + 1 = 0 Zero product property

    r = 2 or r = - 1 Solve each equation

    Solution recurrence relation

    The solution of the recurrence relation is then of the form a_{1} = a_{1 r^n 1} + a_{2 r^n 2} with r_{1} and r_{2} the roots of the characteristic equation.

    a_{n} = a_{1}. 2^n + a_{2}. (-1) "

    Initial conditions:

    1 = a_{0} = a_{1} + a_{2}

    1 = a_{1} = 2a_{1} - a_{2}

    Add the previous two equations

    2 = 3a_{1}

    2/3 = a_{1}

    Determine a_{2} from 1 = a_{1} + a_{2} and a_{1} = 2/3

    a_{2} = 1 - a_{1} = 1 - 2 / 3 = 1/3

    Thus, the solution of recursion relation is a_{n} = 2/3. 2^n + 1/3. (-1) ^n
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