|We need to find,|
|How to get that?|
|Let’s consider the expansion of|
|Now let’s substitute values for|
|if you sum them from top to bottom|
|Taking the sum of arithmetic series,|
Another good resource http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm
If you don’t have a drill guide like http://www.sears.com/shc/s/p_10153_12605_00967173000P?BV_UseBVCookie=Yes&vertical=TOOL&pid=00967173000 then drilling straight using a hand drill will be a pain and will often result angled holes.
Here’s a simple and elegant (though not precise) solution. You will need a square ruler like http://image.made-in-china.com/2f0j00tvREcekzhJbq/Angle-L-Square-Ruler.jpg. The trick is that all drills have visible center line. Use the square ruler to guide this in a straight line.
When you drill without a guide as shown above, the drill is free to move along both blue and red axes. You can restrict this by using it as shown below with the square ruler.
The square ruler has a flat edge, so when you keep it as above, it will stay perpendicular to the drilling surface. You can rotate the drill to have its center line and the square ruler to be in the same vertical plane as blue axis is. Then, with bit of patience, you can guide the drill while keeping the center line in the same plane. Still the drill is free to move in both directions, but you can use the ruler to guide it very well restricting any movement along red axis.
OK, what about movement along blue axis? This is where you need to have some practice. You can use your eyesight to keep the distance between drill bit and the vertical edge of the ruler constant while drilling to make sure you are not moving along blue axis.
As I said earlier, this is not perfect or highly precise, but when you don’t have any the fancy guiding tools, this works like a charm.
Image source for the drill, hand, and wooden piece http://en.wikipedia.org/wiki/File:Drill_scheme.svg