# 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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