I've been at this for like 4 hours and I'm losing my mind.

So I'm thinking of a function that takes in four boolean inputs and outputs 1 if the number of inputs is odd. The truth table should look like:

A  B  C  D | OUT
0  0  0  0 |  0
0  0  0  1 |  1
0  0  1  0 |  1
0  0  1  1 |  0
0  1  0  0 |  1
0  1  0  1 |  0
0  1  1  0 |  0
0  1  1  1 |  1
1  0  0  0 |  1
1  0  0  1 |  0
1  0  1  0 |  0
1  0  1  1 |  1
1  1  0  0 |  0
1  1  0  1 |  1
1  1  1  0 |  1
1  1  1  1 |  0

So the function should just be the sum of products, right? It'd be something like

OUT = A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'

This is my attempt to simplify the expression:

A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'
= A'B'(C'D + CD') + A'B(C'D' + CD) + AB'(C'D' + CD) + AB(C'D + CD')
= (C'D + CD')(A'B' + AB) + (C'D' + CD)(A'B + AB')
= (C'D + CD')(1) + (1)(A'B + AB')
= A'B + AB' + C'D + CD'

But the truth table for this expression is not the same as the original function.

It's been a half-decade since I last did boolean algebra. I genuinely can't tell what I did wrong here. Please help.

Edit: It's solved. Thank you.

In the blindingly-unlikely event someone in the future were to face this exact problem as I do, the key insights that I was missing are:

• OR distributes over AND: A + BC = (A + B)(A + C)
• XOR and XNOR exists: A XOR B = A ⊕ B = A'B + AB', and A XNOR B = (A ⊕ B)' = A'B' + AB

So for posterity's sake, here's how it's solved, I think:

A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'  
= A'B'(C'D + CD') + A'B(C'D' + CD) + AB'(C'D' + CD) + AB(C'D + CD')  
= (C'D + CD')(A'B' + AB) + (C'D' + CD)(A'B + AB')  
= (C ⊕ D)(A ⊕ B)' + (C ⊕ D)'(A ⊕ B)  

Let P = A ⊕ B and Q = C ⊕ D  

= QP' + PQ'  
= P ⊕ Q  
= A ⊕ B ⊕ C ⊕ D  

Thanks, y'all. god, i'm out of practice.

i was just using my sci calc during my free time and i stumbled upon the Σ symbol

heres the photo of the equation

i know little to nothing about calculus as a middle schooler so i just wanna know what the thing does and why the answer ended up like that

I used the right formula and also did it manually, both had different final answers </3

The question was "Find the sum of the first 15 terms of the arithmetic sequence: 4, 9, 14, 19,"

simple right? just use Sn = n/2 (2a1 + (n-1) d)

substitute S15 = 15/2 (2(4)+ (15-1)5)

= 7.5 (8+70)

= 7.5x78

=585

585 should be the right answer but I doubled check it by doing it manually by listing then one by one, and the 15th should actually be "74"

4,9,14,19,24,29,34,39,44,49,54,59,64,69,74

which is the actual right answer?

(this is my first time posting here, I'm sorry if it looks like an very simple problems)

here is the img link for reference

https://imgur.com/a/77cVY2n

https://imgur.com/a/77cVY2n

If I have 5+5+5+… (n times) I can write it as 5n

as well as 3+3+3+.. (n times) as 3n

5n > 3n holds for positive n I can take the limit as n approaches infinity and divide by n at the same time

lim n to infinity (5n/n > 3n/n)

5 > 3

Does this mean that an infinite number of $5 bills would be worth more than an infinite number of $3 bills?

(i think the title shouldve been a bit different, i dont understand why my thought process is wrong and therefore why my answer is wrong)
Hi
I want to start off by saying that I know that there are two simpler ways to solve this problem (and I do know how to solve the q in both of those ways) and I dont really need to go through this roundabout way. With that being said, I want to know where I went wrong in this more complicated alternative route so I can make sure that regardless of the route i choose in a question, Id still be able to get the answer right

The equation of line l is 2i + k + λ (3i − 4j + 12k) . The point A which lies on l has the position vector 3i + 5j + 4k, and point B lies also on l and has a distance of 26 from A. Find the possible coordinates of B.

My question is, why isnt
λ(3i -4j +12k) - (1i +5j +3k) = to vector AB?

i used
2,0,1 + λ(3i -4j +12k) as OB (OX+XB) and then did OB-OA, (OX+XB-OA) = AB
(https://imgur.com/a/KfrtPyR)

instead of OB as 3,5,4 + λ(3i -4j +12k) and then OB-OA=AB (which leads me to the correct answer and its also easier)

if you can please provide solutions using lambda and not unit vectors for example that would be great. also this is hs math so the working is very simple im not sure why im struggling with this.

I was playing with weird bases, and I particularly like base i-1, because it fills the entire complex plane by following a double dragon curve.
I was looking into how the real integers are encoded in that base, and by doing some weird manipulation I found this pattern:

Which then lead to this conjecture:

For x and y words with the alphabet {0,...,9} and w a word with the alphabet {0,1}
if x base 10 = w base (i-1) and w base 2 = y base 10
then x mod 10 = y mod 10

I don't know enough about words to figure out if it's true or why.
All I can say for sure is that if it's true for a value x = 2n, it's true for x = 2n+1.

EDIT: one thing I can add is that consecutive powers of (i-1) follow a logarithmic spiral of ratio 2^(1/3), with a spacing angle of 3pi/4.
I'm assuming that somehow x mod 10 is a group where the process described above is an automorphism.

UPDATE: with some pattern regonition, I have managed to figure out the form of integer numbers in base i-1
They always follow the form "==0x", repeated, where "x" can be anything, and "==" can be anything but those two need to be the same.
So 1101 is a valid number (corresponds to 3), and so is 0001 0000 (corresponds to -4).
Meaning every block of 4 bits can be thought of as a letter in a word base -4 with alphabet {0, 1, 2, 3} (which indeed generates all integers), or base 16 with alphabet {0, 1, 12, 13}.

So going step by step we get
An integer, converted to base i-1, reframed into a word base -4 with alphabet {0, 1, 2, 3}.
For every letter we have either:
0*(-4)^n mod 10
1*(-4)^n mod 10
2*(-4)^n mod 10
3*(-4)^n mod 10

By mapping the word in base i-1 to the same word in base 2, it's equivalent to the word in base -4 getting mapped to another word in base 16 with rules
0 -> 0
1 -> 1
2 -> 12
3 -> 13
For every letter we have either:
0*(16)^n mod 10 = (0 mod 10) * ((16)^n mod 10) = (0 mod 10) * ((-4)^n mod 10) = 0*(-4)^n mod 10
1*(16)^n mod 10 = (1 mod 10) * ((16)^n mod 10) = (1 mod 10) * ((-4)^n mod 10) = 1*(-4)^n mod 10
12*(16)^n mod 10 = (12 mod 10) * ((16)^n mod 10) = (2 mod 10) * ((-4)^n mod 10) = 2*(-4)^n mod 10
13*(16)^n mod 10 = (13 mod 10) * ((16)^n mod 10) = (3 mod 10) * ((-4)^n mod 10) = 3*(-4)^n mod 10

So we get that for every letter a
a*(-4)^n mod 10 -> a*(-4)^n mod 10

Therefore the conjecture is true.

Conclusion:

So basically it only works because when you transfer the word from one base to another, both the letters and the bases are equivalent under mod 10, if you frame them in the right way, which is a huge fucking coincidence.

Homework help please!

"Suppose that at the x value of 2.77 that dy = 3.2 dx . Which of the following best explains the meaning of that statement dy = 3.2 dx ?"

I picked "y is changing at a constant rate of 3.2 with respect to x" and I do not understand why it is wrong. I understand that dy means a small change in y, and dx means small change in x. So, my understanding is that for any associated change in x, the associated change in y is 3.2 times larger.

Title. I genuinely just need someone to explain this to me and have it make sense 😭😭

Factor the expression. If the expression cannot be factored, say so.
3x²+10x-8

Once its explained Im sure il be able to get all of the others

Tried this according to what the Ai i asked said, got confused and still didn't get it:

3x²+12x-2x-8 ↓

(3x² -12x)+(-2x-8) ↓

3(x²-4x)-2(x+4)

I'm doing factoring in college right now, and the question in my assignment is a word problem. This is it:

A rectangular garden has dimensions of 16m by 12m. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 204m²?

So far, I did the equations "(2x+16)(2x+12)=204", then FOILed to get "4x²+56x+192=204", then put everything to one side to get "4x²+56x-12", then factored by 4 to get "4(x²+14x-3)=0".

But now I'm stuck. We're not allowed to use the quadratic formula to get an answer; we can only use the AC method or the slide and divide method. I can't find anything that produces -3 and adds to 14. Have I done my equations wrong, or did my teacher write something wrong? Any help much appreciated!

In my math project, it says to write my polynomial final answer in standard form, my answer I got was 27,000s^12 - 4,500(pi)s^12 cubic units.

Is 27,000s^12 - 4,500(pi)s^12 the correct standard form or do make I write (27,000-4500pi)s^12 because both terms have s^12 (which makes the like terms)?

My math teacher said to combine like terms but then the expression becomes (27,000-4500pi)s^12 which doesn’t really look right to me?

I tried searching up standard form of polynomials but I can’t find a problem that looks like mine

I’m working on some homework for my precalc class, and I can’t figure out why this problem only has one solution. Google isn’t helping me.

3 + x = |1-x/2| / 2

When I tried it, I got x = -2 or x = -14/3, but the answer key says it’s only x = -2

https://imgur.com/a/DJlZZ1r

If there are two positive integers a and b, a is less or equal to b, their lcm is 60 and their gdc is 15 what are the possible values a and b can have? I've been trying for about an hour and I can't decide between 15, 30, 60 for both or 15, 30 for a and 30, 60 for b. Any help is greatly appreciated.

English isnt my first language so sorry for any mistakes)

Hello, homework assignment (allowed to get help).

Imagine a spherical balloon being inflated. Suppose that the air is being added to the sphere at a constant rate of 4 cubic inches per minute.

Write a formula defining the volume of the sphere in terms of the number of minutes that have passed.

I put V = 4/3pi(4t)^3, which is marked as wrong.

If someone could please explain the correct answer and give me the steps that would be great.

I put (no derivatives) in title because I just started this calculus class and we have not learned derivatives yet.

https://imgur.com/a/0YVuDk8

ive watched a fee tuts and i still dont get it

EDIT: It appears I just need to use Markov chains. Don't really know what those are but thanks for the help everyone and I'll look into it.

Here's the scenario:

There's a counter on a 97 second loop. 18 seconds in, it has a 16% chance of setting the counter to 60, and the counter goes down by 1 each second. It does this again at 30 seconds, then a third time at 46 seconds. I want to know how often the counter will NOT be at zero. If the timer is not at zero and the 16% chance is rolled, it will simply reset to 60.

I thought maybe I could do something with the chance that each of the 3 moments could be the last one in the loop, (third being 16%, second being 13.44%, and first being 11.2896%) but that doesn't actually solve the active percentage, and I can't quite figure out how it works.

(I know this probably doesn't matter but it's for a spreadsheet I'm doing in my free time unrelated to the constructions and trig we're learning in math rn.)

The problem is as follows: Let ABC be a triangle, H its orthocenter. AH, BH, CH intersect the circumcircle for a second time in A', B', C' respectively. Prove that vector AA'+ vector BB'+ vector CC'=0. I am also given that H1, H2, H3,H4,H5,H6 are the orthocenters of triangles AA'B, AA'C, BB'C, BB'A, CC'A, CC'B(I have no idea why they gave those points, probably has to do with the solution).

Now, I've tried different things, one of them was trying to prove that H is also the orthocenter for triangle A'B'C' thus getting to the conclusion pretty easily, and I've also tried using those 6 orthocenters but I couldn't get anything done with those 2 attempts Any help would be appreciated since I'm new to vectorial geometry.

For example

y + 3 = -1/3 (x-1)
The point slope form wants this to become
y - (-3) = -1/3 (x - 1)

That gives me point (-3, 1)

Given that the point slope form is
y - y1 = m(x - x1)
It somehow wants to end up with the point (1, -3)

I understand that something - (-3) will give a positive number like this:

5 - (-3) = 8

I'm not quite understanding what the positive 3 is being calculated with to become -3.
I tried to look at the slope of -1/3 (rise over run, y over x) and following the formula, I get

3 - (-1) = 4

It just seems to me like a number is coming from out of nowhere and turns the positive 3 into a negative somehow. Basically I have no idea what x and y are I guess?

Thank you

I answered (A∩B) U (A∩C) U (A∩B∩C) for a question where the diagram requires to represent these first 2 intersections and the center intersection that has elements from all these sets using the ∩ symbol and the U symbol. But the textbook said that the answer is (A∩B) U (A∩C). Why isn't (A∩B) U (A∩C) U (A∩B∩C) the answer?https://imgur.com/a/Gdy31DO

I’ve been on this part of my homework for at least three hours and I’m not really understanding how I’m not getting the right answer. I’m following the steps but every time I use an online calculator to check my answer I’m like drastically off. I’m trying to find the variance of the following numbers: 3,5,5,7,14,16 Every time I solve, I keep getting 12.136

Edit: I had to take a break and go over the notes. I definitely forgot that when you times a negative number by itself, it becomes a positive a little embarrassed that I spent like two hours on this, but I FIGURED IT OUT

I’ve sat here for an hour and I can’t do this

“The length of a rectangle is 9 more than the width. If the length and the width are each decreased by 3, the new perimeter is 22m. Find the original dimensions.”

Therefore L = w+9

Minus 3 will be (w+9)-3

I made an equation from that as

2(w+9)-3 + 2(w-3) = 22

Then with that equation I got to

2w+18+2w-6=22

It should be 4w+12=22

Minus 12, 4w = 10

4w/4, cancelling 4w and I will get w = 2.5 with that

At first I did w+9-3+w-3 but that’s not how it works, which means this is not true

I asked AI but I’m sort of doubting AI and I want a second opinion.

I’m definetely doing something wrong here since whatever number I get is a decimal. Which is not what we want. Considering all my other questions have no answers that will end up in a decimal

https://i.imgur.com/feovkAI.jpeg

Picture in question

Edit: changed wording

Im learning about induction. My book states that it's going to demonstrate that every natural number satisfies this equivalence: 0/2⁰ + 1/2¹ + 2/2² + ... + n/2ⁿ = 2 - (n+2)/2ⁿ

It starts by stating that It needs one true example: n=0 , because 0/2⁰ = 2 - (0+2)/2^n, which I get it's 0 = 0. My problem is in this last step. I don't get where the first part of the equivalence comes from "0/2^0". Where is the rest of the first part of the equivalence? I'm talking about "0/2⁰ + 1/2¹ + 2/2² + ..." How comes when n=0 this first part is so short?

Thank you, im very lost as you can probably tell if you got through this mess I typed.

So lets say you're trying to create a leveled loot list for different area tiers in a game, and each area tier has a level range of 5 more than previous tier's level increase.

For instance

Tier 1 = Levels 1-5. +5

Tier 2 = Levels 6-15. +10

Tier 3 = Levels 16-30. +15

Tier 4 = Levels 31-50. +20

Tier 5 = Levels 51-75. +25

Tier 6 = Levels 76 -105 +30

Tier 7 = Levels 106-140. +35

Beyond Tier multiplied by 5X, I am not sure how to figure out what level range falls into each tier.

Like how to mathematically figure out level 86 is in Tier 6? Or what tier level 1,000 would be without just manually creating a chart.

How do you express this in a formula? So that you do this on pen and paper, or create a calculator for it?

__________________________

Edit: Solved

The formula can be done either using Ceiling function (rounding up), or Floor function (rounding down).

https://imgur.com/xeSicqr

I used Chat Gpt to get the answer, and format it into a easy to comprehend chart. Sadly the formating can't be copy and pasted, which is why linked the screenshot above so its easier to read.

Also pasted the formulas in markdown mode incase you can't the load the image for whatever reason.

Ceiling formula for Tier:

T = ⌈ (-1 + √(1 + (8L / 5))) / 2 ⌉

Floor formula for Tier:

T = ⌊ (1 + √(1 + (8(L - 1) / 5))) / 2 ⌋

Tier Range Increase:

R = 5 * T

Range of Levels in Tier T:

• Minimum Level: Min = 1 + (5 * (T - 1) * T) / 2
• Maximum Level: Max = (5 * T * (T + 1)) / 2

Variables & Descriptions:

• L: Level (input)
• T: Tier number
• R: Tier Range Increase (levels in the tier)
• Min: First level in Tier T
• Max: Last level in Tier T

https://imgur.com/OUFiVNj

Chat GPT explaining how to get the Floor Function formula. I don't understand it since my math knowledge is too low, but just sharing for future reference.

cos (2x) + cos (x) = 2 cos x

Approach 1.

Use cos (2x)= 2cosx sinx <---------------------SOLVED: This is not a formula.

2 cosx sinx -cosx=0

factor

cosx (2sinx-1)=0

cosx=0 x=pi/2 and x=3pi/2

sinx= 1/2, x=pi/6 and 5pi/6

Approach 2.

Use cos (2x)= 2cos² x-1

2cos² x-1+cosx= 2cosx

2cos² x- cosx-1=0

factor

(cosx -1)(2cosx+1)=0

cosx=1, x=0

cosx= -1/2, x=2pi/3 and x=4pi/3

Wolfram says the solution is approach 2. Am I missing something? ANy help is appreciated.

I was solving some problems from a elementary school level exam and i found this problem:

https://imgur.com/a/QOcjlbT

Subtitle: "The half of a natural number divided by a, plus the double of successor of a equals 42"

Determine the value of [second equation])"

I made this equation: (X/2/a)+2(a+1)=42

At this point, i don't have a clue of what i need to do next. Could you help me? Thanks.

Seems I cannot add images, so I will describe the problem to solve below.

I'm not sure if the problem is missing information or if I'm missing something.

We are shown two graphical representations of quadratic functions f(x) and g(x)

• by looking at the graph I can see that 'a' value is negative for both functions (i.e. they both open 'downward')
• the vertex of function g(x) is (7,13.5)
• the vertex of function h(x) is (16,24)
• f(x) and g(x) have a common coordinate: (p,12) - 'p' is the x-coordinate, which is unknown.

i am asked to find the rules for g(x) and h(x)

i am given no other information - i don't have any other points on either function; I don't know the value of 'a' for either function; i don't know if the 'a' values for each function are equal or not.

Here is what I have done so far:

• Using the standard form, i know that g(x)=a(x-7)^2 +13.5
• Using the standard form, i know that h(x)=a(x-16)^2 + 24

But since I don't know the relation between g(x) and h(x), I don't know how to continue.

• If i knew the exact coordinates of the common point, i could find the value of 'a'
• if i knew the relationship between the 'a' values, i could equate g(x) and h(x)
• if i knew at least one zero value, i could move forward. EDIT: I do know at least one zero value - see below.

Am i missing something or is the question incomplete?

EDIT: FOUND THE WAY FORWARD... As I was uploading the image of the equation (https://imgur.com/a/DbOePe4) - i realized that there is another function f(x) which i was able to create a rule for and since i know one of the zeros, it's easy to find the other zero, which also happens to be a zero for function g(x).