What is the square root of the sum of the first 2007 positive odd integers?
$\displaystyle \sqrt{\sum^{2007}_{n=1}(2n-1)}$
$\displaystyle =\left(2\sum^{2007}_{n=1}(n)-\sum^{2007}_{n=1}(1)\right)^\frac{1}{2}$
$\displaystyle =\left(2\left(\frac{2007(2007+1)}{2}\right)-(2007)\right)^\frac{1}{2}$
$\displaystyle =\left(2007(2007+1-1)\right)^\frac{1}{2}$
$\displaystyle =\left((2007)^2\right)^\frac{1}{2}$
$\displaystyle =2007$
Hello, Rimas!
This could be a trick question . . .
We have: .$\displaystyle S \;= \;\underbrace{1 + 3 + 5 + 7 + \cdots }_{2007\text{ terms}}$What is the square root of the sum of the first 2007 positive odd integers?
This is an arithmetic series with first term $\displaystyle a = 1$ and common difference $\displaystyle d = 2$
The sum of the first $\displaystyle n$ terms is: .$\displaystyle \frac{n}{2}[2a + (n-1)d]$
So we have: .$\displaystyle S \;=\;\frac{2007}{2}[2\cdot1 + 2006(2)] \;=\;2007^2$
Therefore: .$\displaystyle \sqrt{S}\;=\;\sqrt{2007^2} \;=\;2007$
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If you knew the following bit of mathematical trivia,
. . you could have answered it immediately.
$\displaystyle \begin{array}{ccccc}1 & = & 1 & = & 1^2 \\1 + 3 & = & 4 & = & 2^2 \\1 + 3 + 5 & = & 9 & = & 3^2 \\1 + 3 + 5 + 7 & = & 16 & = & 4^2 \\1 + 3 + 5 + 7 + 9 & = & 25 & = & 5^2\end{array}$
Get it?
The sum of the first $\displaystyle n$ odd number is always $\displaystyle n^2$.
So the sum of the first 2007 odd number is $\displaystyle 2007^2$.
And its square root is, of course, $\displaystyle 2007$.
Hello, ecMathGeek!
A straight-forward, no-nonsense approach . . . great job!
I intended to use your method, but something bothered me.
The square root of the sum? . . . Won't that be some ugly decimal?
. . Why in the world would they want such a . . . Oh!
I remembered that bit of trivia . . . and I was literally LOL.
Then I had to devise an approach . . . and I used Arithmetic Series.
Thank you.
I started it thinking the same thing (but figured/hoped it would work out in the end). Just after I set up the summation, I remembered that same trivia that you spoke of and knew at that point what the answer would be. So, relieved that there would be a solution, I persevered until I reached it.