WEI L asked in Science & MathematicsMathematics ยท 1 decade ago

Mathematical Induction?

I understand how to do regular problems using mathematical induction, but now my teacher assigned us harder problems that we haven't learned yet.

Use mathematical induction to prove that n < 2^n whenever n is a positive integer.

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  • 1 decade ago
    Best Answer

    First prove the base case, n=1: 1 < 2^1 = 2 is true.

    Now, suppose that n < 2^n. We need to prove that n+1 < 2^(n+1).

    We know n < 2^n, so n + 1 < 2^n + 1

    < 2^n + 2^n (since for n positive 2^n > 1)

    = 2^(n+1)

    So n+1 < 2^(n+1) also.

    So, by induction, n < 2^n for every positive integer n.

  • 1 decade ago

    For n=1: n<2^n {1<2}

    For n=2: n<2^n {2<4}

    Assuming P(n) = n, hence P(n) < 2^n

    Now, we need to prove that P(n+1) < 2^(n+1).

    P(n+1) = n+1 <2^n + 1

    Considering R.H.S:

    2^n + 1 < 2^n.2 as 2^n > 1 (for n>0).

    Hence, P(n+1) < 2^(n+1)

    Proved.

  • 1 decade ago

    n < 2^n

    lim. n =>0

    0>1

    let n be e

    e< 2^e

    ln e < e ln2

    1 < e ln2

    now, let n be 1

    1 < 2^1

    1 < 2

    It is now possible to induce that in all cases where n is an integer that n will be smaller than 2^n

  • 1 decade ago

    prove n=1 is easy. (1 < 2^(1) is true)

    assume n=k is true, then

    Left-Hand-Side

    = k+1

    < 2^k + 1 ......................( k < 2^k is true)

    <2^k +2^k ......................( 2^k > 1 when k > 1)

    =2 (2^k)

    =2^(k+1)

    =Right-Hand-Side

    so, k+1 < 2^(k+1)

    Hence, we know that when it is true for n=k, it is true for n=k+1

    By mathematical induction, it is true for n >= 1 (positive integer)

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  • Anonymous
    1 decade ago

    assume n=k is true, then

    Left-Hand-Side

    = k+1

    < 2^k + 1 ......................( k < 2^k is true)

    <2^k +2^k ......................( 2^k > 1 when k > 1)

    =2 (2^k)

    =2^(k+1)

    =Right-Hand-Side

    so, k+1 < 2^(k+1)

    Hence, we know that when it is true for n=k, it is true for n=k+1

  • 1 decade ago

    Its a useless chapter.

  • 1 decade ago

    you must have mr. sutton ..lol

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