CoolStuffYT

If the sum of the last 5 terms is 85, and there are 10 terms, then the eighth one is 17 (take 85/5).

And if the fourth term is 9, then the common difference is 2 (pretty simple math).

To find the sum, we just find the median and multiply it by the number of terms.

The median is 12, multiplied by 10, gets us 120.

You are very welcome!

:P

Let's say the coin landed on heads, which is a 1/2 chance.

Then the chance of it being red is 4/6 = 2/3.

Therefore, the total so far is 1/2*2/3 = 1/3

Now if the coin landed on tails, 1/2 chance

Red is 2/6 = 1/3

1/2*1/3 = 1/6

Total probability - 1/3 + 1/6 = 1/2

You are very welcome!

:P

Hey calcuser

This is \( 101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2.\)

And \((a +(2n+1)d)^2- (a + (2n)d)^2 +(a + (2n-1)d)^2 - (a+(2n-2)d)^2 + \cdots + (a+d)^2 - a^2.\)

I'm not sure how to solve it but I DO know how to put it into LaTeX...

1+2-3-4 = -4

5+6-7-8 = -4

And so on. 

Find the number of -4's

Let's call the smallest number x

So x+x+1+x+2+x+3+x+4 = 5x+10 = A

Then x+5+x+6+x+7+x+8+x+9 = 5x+35 = A+25

You are very welcome!

:P

10+12+14+...+498+500

Pair these up

500+10 is a group which equals 510

498+12 is a group which equals 510

Then find how many 510 groups there are, and mulitply that by 510

\(x=\frac{3\pm \sqrt{9-4\times 2\times 27}}{4}\)

\(x=\frac{3+3\sqrt{23}i}{4}, \frac{3-3\sqrt{23}i}{4}\)

Square both to get

\(x^2=\frac{-198+12\sqrt 23i}{16}, \frac{-198-12\sqrt 23i}{16} \)

Now just add them up.

You are very welcome!

:P