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Asked: 2026-05-26T00:38:36+00:00

I am trying to understand a proof by induction in my algorithms text book.

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I am trying to understand a proof by induction in my algorithms text book. Here’s the author is proving using induction that the T(n) will always be greater than 2^(n/2) (This is for calculating the nth fibonacci number using the recursive algorithm):
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What I don’t understand is the very last step, where he is manipulating the equation. How does he go from:

> 2^(n-1)/2 + 2^(n-2)/2 +1

to

> 2^(n-2)/2 + 2^(n-2)/2 +1

He just randomly changes 2^(n-1)/2 to 2^(n-2)/2. Is this a mistake?

Thanks.

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1 Answer

  1. Editorial Team

    I believe that particular step runs off the assumption that:

    T(n-1) > T(n-2)

    Therefore, we can form an algebraic inequality:

    T(n-1) + T(n-2) + 1 > T(n-2) + T(n-2) + 1

    We can lop off the + 1 from the right side (because the inequality will still hold true for anything subtracted away on the LESSER side):

    T(n-1) + T(n-2) + 1 > T(n-2) + T(n-2)

    From this, substitute our T(m) = 2^(m/2) (for anything less than n and > 2, which n-1 and n-2 both qualify):

    2^(n-1)/2 + 2^(n-2)/2 + 1 > 2^(n-2)/2 + 2^(n-2)/2

    That gets you that particular step. It’s done this way deliberately as the poster above me stated, to get to 2^(n/2).

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