1. ## smallest integer inequality

find the smallest positive integer n satisfying the inequality: 1+2+3+....+n>2007

whats an inequality?

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2. Originally Posted by mich13
find the smallest positive integer n satisfying the inequality: 1+2+3+....+n>2007

whats an inequality?

xxxxxxxx

$\displaystyle \frac{n(n+1)}{2} > 2007$

Now solve.

3. Hello, mich13!

Find the smallest positive integer n satisfying the inequality: .$\displaystyle 1 + 2 + 3 + \cdots + n\;>\;2007$

whats an inequality? . . . . You don't know?
The symbol $\displaystyle {\bf{\color{blue}>}}$ means "greater than".

You may be expected to know this formula . . .

The sum of the positive integers from $\displaystyle 1$ to $\displaystyle n$ is: .$\displaystyle \frac{n(n+1)}{2}$

So we have: .$\displaystyle \frac{n(n+1)}{2}\;> \;2007\quad\Rightarrow\quad n^2 + n \;>\;4014$

We have a quadratic inequality: .$\displaystyle n^2 + n - 4014\;>\:0$

The equation: .$\displaystyle n^2 + n - 4014 \:=\:0$

. . has roots: .$\displaystyle n \;=\;\frac{-1 \pm\sqrt{1^2 - 4(1)(4014)}}{2(1)}\;=\;\frac{-1\pm\sqrt{16057}}{2} \;=\;\begin{Bmatrix}\;62.658... \\ \text{-}64.739...\end{Bmatrix}$

Therefore: . $\displaystyle \boxed{n\:\geq\:63}$

4. ## thanks you but....

Originally Posted by Soroban
Hello, mich13!

The symbol $\displaystyle {\bf{\color{blue}>}}$ means "greater than".

You may be expected to know this formula . . .

The sum of the positive integers from $\displaystyle 1$ to $\displaystyle n$ is: .$\displaystyle \frac{n(n+1)}{2}$

So we have: .$\displaystyle \frac{n(n+1)}{2}\;> \;2007\quad\Rightarrow\quad n^2 + n \;>\;4014$

We have a quadratic inequality: .$\displaystyle n^2 + n - 4014\;>\:0$

The equation: .$\displaystyle n^2 + n - 4014 \:=\:0$

. . has roots: .$\displaystyle n \;=\;\frac{-1 \pm\sqrt{1^2 - 4(1)(4014)}}{2(1)}\;=\;\frac{-1\pm\sqrt{16057}}{2} \;=\;\begin{Bmatrix}\;62.658... \\ \text{-}64.739...\end{Bmatrix}$

Therefore: . $\displaystyle \boxed{n\:\geq\:63}$

no i dont know! I am 13 years old and i have no idea how to solve this!! i found it difficult to understand your meathod, but i managed! how did you get 4014 though? please explain it! thanks a lot xxx

5. Originally Posted by mich13
no i dont know! I am 13 years old and i have no idea how to solve this!! i found it difficult to understand your meathod, but i managed! how did you get 4014 though? please explain it! thanks a lot xxx
2007 x 2 = 4014