uvacowdo

thanks so much!

im tryna upvote your answers but it doesnt work! 

thank you!

if this isnt too much to ask...could you try the others?

hehe... i put them now

do you think you could include the grid?

Since the triangles are similar, that means the ratio between each of the sides will be same.

You can find that the ratio between sides is 2:7 (by comparing AB to XY and BC to YZ).

Since we're tryna find N, you need to divide AC by two (to get 1.5) then multiply by 7.

The answer is 10.5

Sorry,,, \(f(x) = \left\lceil\dfrac{1}{x+2}\right\rceil$ for $x > -2$, and $f(x) = \left\lfloor\dfrac{1}{x+2}\right\rfloor$ for $x < -2$. ($f(x)$ is not defined at $x = -2\)

Looking at the multiple choices, you can immediately rule out the first two, because the exponent is only applied to the fraction in the parentheses. Then, looking at the remaining options, it is the third one because when you raise a fraction to a certain power you raise both the numerator and the denominator to that power. So, \((\frac{1}{4})^2 = (\frac{1^2}{4^2}) = \frac{1}{16}\) so the answer is the third option.

You can start by doing 6x6, which is 36. If you multiply it by 6 again, the units digit is still six. So, the last digit is always going to be 6.

6^1 = 6

--------------- the stuff in between the two dash lines is one cycle of the pattern

6^2 = 36

6^3 = 216

6^4 = 1296 (by the way, for these you don't have to multiply the whole thing out, you just need to multiply the last two digits of each result by 6 until you see a pattern lol)

6^5 = 7776

6^6 = 46656

-----------------

6^7 = 279936

And yay! We've arrived back at the start of the pattern, with the tens digit being 3. There's 99 numbers from 2-100, and 99/5 is 19R4 and the 4th tens digit in the pattern cycle is 7.

So the last two digits are 7 and 6.

This thing (the problem lol) is an arithmetic progression.

there's a formula for this type of problem.

Let's say A is the first term and A_n is the last term, and d is how much each number is increasing from the last. so first we should first find the number of terms in the progression. the equation is:

A_n = a+(n-1)d

so ya plug in the values:

99=1+(n-1)2

98=(n-1)2

n-1=49

And n=50

There are 50 terms in all…

Now the sum of these fifty terms:

S_n = n/2(A+A_n)

S₅₀=25(1+99)

Sooooo the answer is 25*100 = 2500

First, we should prime factorize all the numbers:

21 = 3x7

35 = 5x7

60 = 3x4x5

The GCD is just the product of all the factors that all three of the numbers share. Since there's no common factor for all three of them, the answer is 1. (because the 1 is a common divisor for all real numbers) 

:)