Math - PreCalc (12th Grade)

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What is the first natural number that causes the statement 1 + 3 + 5 +... + (2n − 1)<=4n − 1 to fail?

A) 2
B) 3
C) 4
D) 5
E) 6

  • Math - PreCalc (12th Grade) -

    start checking. How about showing some of your own ideas on further problems? I can do them - the goal is for you to work on them, eh?

  • Math - PreCalc (12th Grade) -

    Honestly, i don't get any of them..

  • Math - PreCalc (12th Grade) -

    it asks for the first natural number ...
    the natural numbers are 1,2,3,...

    So, start trying them:

    n=1: (2n-1) = 1
    1 <= 4*1-1
    1 <= 3? yes

    n=2: (2n-1) = 3
    1+3 <= 4*2-1
    4 <= 7? yes

    n=3: (2n-1) = 5
    1+3+5 <= 4*3-1
    9 <= 11? yes

    n=4: (2n-1) = 7
    1+3+5+7 <= 4*4-1
    16 <= 15? NO

    Or, you can solve the inequality. As you can see,

    1+3+5+...+(2n-1) = n^2

    So, we want to see where

    x^2 <= 4n-1
    x^2 - 4n + 1 <= 0

    this parabola has a root at 2+√3 = 3.7
    So, as long as x <= 3.7, x^2 <= 4n-1
    The first natural number greater than 3.7 is 4.

  • Math - PreCalc (12th Grade) -

    Thank you!! you really are helping me alot.

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