PartialMathematician

It was discovered because some ancient person wanted to count how many berries were on a tree.

You can learn the binomial theorem and memorized pascal's triangle to help yourself.

- PM

n choose 1 is always n. Similarly, n choose n and n choose 0 are always 1. 

 \({n}\choose{1}\) = n. \({n}\choose{n}\) = \({n}\choose{0}\) = 1.

- PM

I don't think you are done with your answer, so I'll just finish it for you. 

Substituting into b = a - 8 and c = a - 5 is probably more helpful...

We have the equation a+b+c=221, and we understand that \(b = a-8\) and \(c= a-5\). Substituting b and c with their values in terms of a, we have \(a + a - 8 + a - 5 = 221, \), which simplifies into \(3a = 234\). Dividing both sides by 3, we have  \(a = 78\) , so a's height is \(\boxed{78}\) inches. 

If you wanted to know the height of b and c, we can revisit the equations b = a - 8 and c = a - 5. Simplifying b = 78 - 8, we know that b is 70 inches tall, and simplifying c = 78 - 5, we know that c is 73 inches tall. 

If you want to check your work, we can add 78, 70, and 73 to get 221, so our answers are correct.

Hope this helps, 

- PM

If you think about it, there are 100 possible intersections of points on the square where the centroid could be, but the points cannot be collinear, so the centroid cannot be on the edge of the square. There are 36 points on the edge of the square, 10+10+8+8 or 9*4, so 100 - 36 = 64. There are \(\boxed{64}\) possible centroids.

- PM 

Thanks, CPhill, both of our answers used the same steps. They were both really good!

Note: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1.

Different powers of i still have the same properties. You can divide the power by 4 and determine the remainder to see if it equals i^1. i^2, i^3, or i^4.

As you can see, i - 1 -i + 1 cancel out each other, equalling 0. The last three terms of your sequence are i^97, i^98, and i^99, which are the same as i^1, i^2, and i^3. i - 1 -i = -1, so the answer is \(\boxed{-1}\). 

Hope this helps, (CPhill will give a more vivid answer)

- PM