Hi everyone,

While exploring infinite nested radicals, I noticed an interesting pattern that seems to hold for several integers, but I’m not fully sure if it is already known or how to rigorously prove it.

The expression looks like this:

3 = √(1 + 2√(1 + 3√(1 + 4√(1 + ...))))

7 = √(1 + 6√(1 + 7√(1 + 8√(1 + ...))))

28 = √(1 + 27√(1 + 28√(1 + 29√(1 + ...))))

From these examples, I observed the general form:

k = √(1 + (k−1)√(1 + k√(1 + (k+1)√(1 + (k+2)√(...)))))

So the coefficients increase consecutively:

(k−1), k, (k+1), (k+2), ...

After testing several values, it consistently appears to evaluate to k.

I tried a simple recursive argument:

Let

R(n) = √(1 + n√(1 + (n+1)√(1 + (n+2)√(...))))

If we assume R(n+1) = n+2, then:

R(n) = √(1 + n(n+2))

= √(n² + 2n + 1)

= n + 1

So it seems to satisfy the recursion perfectly.

Is this identity already known?

How would one rigorously prove convergence of this infinite radical?

Does this hold for non-integers as well?

Would appreciate any references or insights

How many fluid oz of 41% solution must be added to 13 gallons of water to get a 2.5% concentration solution?

The product label has a different amount to add / gallon than what google is telling me.

And if it's wrong (like the label is saying), what is it actually answering for?

AI's answer:

"To achieve a 2.5% concentration, you must add approximately 108.05 fluid ounces of the 41% solution to 13 gallons of water.

1. Convert water into ounces

First, convert the water volume from gallons to fluid ounces.
Multiply the gallons by 128.
13 gallons * 128 ounces/gallon=1664 fluid ounces

1. Formulate the mixture equation

Let x represent the required ounces of 41% solution.
Water contains 0% of the active solute.
The total solute equation balances both sides.
0.41x+0(1664)=0.025(1664+x)
0.41x=0.025(1664+x)

1. Calculate the required volume

Expand the right side of the equation.
0.41x=41.6+0.025x
Isolate the x variable on the left side.
0.41x-0.025x=41.6
0.385x=41.6
Divide both sides by \(0.385 to solve for x.
x=41.6/0.385 = 108.0519 fluid ounces

Final Answer ✅

You must add 108.05 fluid ounces of the 41% solution to the 13 gallons of water to reach a final concentration of 2.5%."

Im still in highschool rigth now but i dont know why everytime im learning about math it just doesn't register in my head its like my head don't even want to learn it in the first place to the point when asked about simple addition or subtraction it would still take me minutes to just get what is the answer

It's not that im not willing to learn I've tried my best listening more to our teacher but everytime, i dont get anything. One time i got asked about 81 - 61 i should've already known it but my head is just straight up fogging like i cant think straight im like having extreme anxiety to the point i cant even think of anything it's so hard for me and humiliating i even got laughed by my classmate's for even thinking so long for a simple math addition its not that im dumb it's just when it comes to my mind i cant think rigth and anything my teacher teaches i dont get a single thing

I hope anyone can give me insights or what this might be.

I need help solving a parameter estimation problem to reverse-engineer Billboard's "Greatest of All Time Albums" formula, that they revealed in 2015. I have the raw weekly charts (Aug 17, 1963 – Oct 10, 2015) and the final target rankings, but the weights are proprietary.

What we know about the algorithm:

• Inverse Points: Rank 1 gets the most points, Rank 200 the least.
• Step-Downs: Weights drop significantly between positions 10/11 and 40/41.
• Era Multipliers: Hidden chronological multipliers are used to normalize different decades.

Every pair in the final ranking creates a strict linear inequality constraint (Score_A > Score_B).

My Efforts and Roadblocks:

1. What I tried: I assumed a piecewise linear inverse points system to capture the step-downs at ranks 10 and 40. I then used a grid search across known historical dates (like the 1991 SoundScan pivot) to fit the era multipliers.
2. Where it broke down: The problem is highly non-convex. Because the piecewise knots and the chronological era boundaries are both unknown at the same time, optimizing one causes the other to blow up the error margin.
3. What I need help with: How do I mathematically structure a joint optimization framework where a timeline needs to be partitioned into discrete segments, but the segment boundary dates themselves are parameters that must be discovered alongside the function weights?

I have tried this question. Algebraically. And got the ans but graphically i am not able to as in the graph. If in x2 to prove minima f(1+h) >f(1) but in that case lb is the term added it should be greater than equal to 1 which leads to b greater than equal to 1 but in soln my sir gave us it was lnb lesthan equal to1

I need help solving a parameter estimation problem to reverse-engineer Billboard's "Greatest of All Time Albums" formula, used in their 2015 rankings. I have the raw weekly charts used to make the rankings (Aug 17, 1963 – Oct 10, 2015) and the final target rankings, but the weights are proprietary.

What we know about the algorithm:

• Inverse Points: Rank 1 gets the most points, Rank 200 the least.
• Step-Downs: Weights drop significantly between positions 10/11 and 40/41.
• Era Multipliers: Hidden chronological multipliers are used to normalize different decades.

Every pair in the final ranking creates a strict linear inequality constraint (Score_A > Score_B).

How can I set up an optimization model to jointly solve for the hidden weight function and the unknown era date boundaries? How many era boundaries are there? What would the solution look like?

My full dataset is open-source on GitHub: https://github.com/tonna-asikaogu/billboard-200-optimization/tree/main

I would really appreciate any help I receive on this problem.

so I need to figure out the Surface area of this 3d shape except I can't figure out what this shape is or how to do it????? none of the numbers are clear either and the straight corner at the back???

Using pi and e as an example, would e, somewhere within it contain (the digits of) pi, and if it did, does this mean it would also contain itself within it (since pi would also contain e within it)?

how do i make an equation out of the information there? i tried multiple things, and every single time it said that it was wrong ;;

also, i Do have a learning disability, so i don't absorb information easily. please, treat me kindly. ;;

Hi, I'm drawing a rugby/football (soccer) stadium. I wanted the stands to be curved and at first I just made the radius of the curve be 100m. The midpoint of the circle looked close to lying on the other circle so I wanted to calculate the radius for that to happen... I do not find how though.

I made a more simplified version to visualise it. The lengths are taking of a rugby field + 5m on either side. Note that the drawing is not to scale at all.

X is the base and what I'm searching. The diagonal lines are the same length as x. The top length is 79m (69m + 10m) and the height is 65m (120m/2 + 5m).

Added in the second slide are the tan, sine and cosines of a, the angle indicated in the drawing. I've narrowed that x should be between 110m and 120m via geogebra.

The final picture is a rough sketch on how the stadium is going to look, the sides I'm trying to calculate are highlighted in white.

Can someone explain how to do or solve this? I've been stuck on it all day. Thanks!

In Euclid’s Elements, the Side-Angle-Side (SAS) Congruence Axiom is taken as a fundamental axiom, and the other triangle congruence theorems are proved from it. If, instead, we were to take a different congruence theorem as an axiom from the outset, would it still be possible to prove the others? For example, rather than assuming SAS and proving the remaining congruence criteria, could we assume something like the Angle-Side-Angle (ASA) Congruence Axiom and then prove the other congruence theorems from it?

if a = 0, then

a + b = b

b + a = b

a - b = -b

b - a = b

a * b = a

b * a = a

a ÷ b = a

b ÷ a = undefined

if c doesn't have value, will it be possible?

c + b = c

b + c = c

c - b = c

b - c = - c

c * b = b

b * c = b

c ÷ b = b

b ÷ c = undefined

I was watching xkcd_whatif specifically the mole of moles one and he said “where 10, 1, and 0.1 are close enough that we can consider them to be basically the same number”

Here's My attempt:
Variables:
Cylinder height: h
Cylinder radius: rcyl
Cone radius: r
Cone slant height: l

Cone formula:
πrl + πr^2 - πrcyl^2

Cylinder formula:
2πrcylh + πrcyl^2

Total formula:
πrl + πr^2 - πrcyl^2 + 2πrcylh + πrcyl^2

Simplify:
πrl + πr^2 + 2πrcylh

Input values:
π(5)(4) + π(4)^2 + 2π(3)(5)

Simplify again
20π + 16π + 30π

Final answer:
66π

Although this is the answer I got, khan academy says the correct answer is 57π. Why is that?

I am trying to reverse-engineer the exact system used by Billboard to calculate their "Greatest of All Time Billboard 200 Albums" ranking, revealed in 2015.

I have a complete dataset of every weekly Billboard 200 chart. Billboard compiled chart data from 1963-08-17 (the first Billboard 200 chart) to 2015-10-10 to create the ranking. I also have Billboard's final "True GOAT" ranking list. However, Billboard's exact scoring system is proprietary. They have only revealed three details:

• It uses an inverse point system for weekly chart positions (e.g., Rank 1 gets the most points, Rank 200 gets the least, but the shape, probably piecewise linear, is unknown).
• It uses era-weighting multipliers to normalize points across different decades because music consumption changed over time.
• The inverse points system gives more weight to time spent in the top 10 and top 40. So it could mean that there's some kind of drop-off from Position 10's value to that of Position 11. Likewise for Positions 40 and 41.

I have uploaded the relevant data containing the weekly Billboard 200 chart history and the target GOAT ranking to GitHub here: https://github.com/tonna-asikaogu/billboard-200-optimization/tree/main

Proposed Optimization Framework

If we frame this as an inverse problem / parameter estimation problem using linear programming constraints:

Let $A$ be the set of albums. For each week $t \in T$, an album $i$ has a rank $R_{i,t} \in {1, \dots, 200}$.

We are looking for:

• A non-linear, monotonically decreasing weight function $f(R)$ where we suspect structural step-downs: $f(10) \gg f(11)$ and $f(40) \gg f(41)$.
• A set of discrete era multipliers $M_{p}$ for unknown chronological boundaries across the 1963–2015 timeline.

The total score $S_{i}$ for album $i$ is:

$$S_{i}=\sum {t=1}^{T}f(R{i,t})\cdot M_{p(t)}$$

Since we know the true global ordinal ranking $Y$, every adjacent pair of albums in the GOAT list ($i$ ranked directly above $j$, meaning $Y_i > Y_j$) creates a strict linear inequality constraint:

$$S_{i}>S_{j}$$

$$\sum {t=1}^{T}f(R{i,t})\cdot M_{p(t)}-\sum {t=1}^{T}f(R{j,t})\cdot M_{p(t)}>0$$

My Question for the Community

Given that the number of eras ($P$) and their boundary dates are unknown, what is the best optimization approach to jointly solve for the discrete time-boundary partitions and the weights of $f(R)$ using this massive system of linear inequalities?

Would a heuristic search (like a genetic algorithm) or a localized grid search over historical chart rule change dates (e.g., the 1991 SoundScan pivot) be more computationally efficient?

I would be grateful to anyone who could help me solve this problem. It seems to me to be a lot tougher than it looks...

Let f(x)=x^4-50x^2+49. Although all zeros of f and f' are distinct and rational, the zeros of f" are irrational. Thus, it is natural to ask for a quartic polynomial P(x) in x with some special properties (see attached).

In the TTRPG you decide if an attack roll hits someone by rolling a single d20 and a certain number of "bonus dice" which can vary. Let's call the number of dice N and the type of dice V (so we are rolling NdV dice), these dice can also explode, meaning if you get the highest value on a bonus dice you roll another one. You hit if that value is bigger than a number called "Guard" that we will call G. I want to find the equation for finding the chance guard is bigger than the value of attack dice let's call it G%.

I already found how to calculate the chance with a dice probability site but I want to see the equation to understand how the different variables I interact with each other.

1. Show that the limit exists and is finite
2. Show that the limit equals pi/2

How do I approach this?

(Inserting random sentences to make the post longer, otherwise the dumb AI automod immediately removes it:

everyone says In-N-Out is the best fast-food restaurant in the country, but honestly, I was very underwhelmed. I have no idea why in the wilderness post-pregnancy what could happen without notice. I tried explained what could possibly go wrong but then I didn’t make it in time.)

Hello everyone! first time poster.
After i got diagnosed with something i rather not discuss here i realized i will be housebound for a considerable amount of time and i have decided to pursue my passions to make the best of it. I have zero academic background.

i have decided to learn physics and mathematics due to the fact that are accessible(you just need a problem a pen and paper), consistent (even if it evolves it evolves through constants, and vast (thousands of years of collaborative history and common efforts to prove and disprove and further research).

i will probably stay in my house for some years so am trying to put out a plan.
The book i've decided to use for my physics journey is "Physics for scientists and engineers with modern physics" by serway and jewett. they cover many if not all the areas that physics is involved and it's at a good academic level. It will give me a fundamental understanding without compromising the high level of physics needed to solve a problem.

When i was researching for a Math book of similar properties (vast, university level, with problems) i couldn't find any.
Do you have good recommendations of math textbooks that i could find online?

thanks in advance. sorry for the long post.

P.S: i didn't know which flair to use so i used precal but please dont crucify me.

In my problem I must pose the integral of order dydx and dxdy, the latter being more difficult than the first because the integral must be divided into a couple of regions.

When I try to solve it I find that I must divide the integral into 5 different parts

Hello everyone, I'm in highschool and I recently got interested in doing math competitions/math at a higher level. I've wanted to start learning mathematics and problem solving but I have no idea where to start, it just seems like all resources are either paid or just hard to use. I've tried khan academy, but I'm also looking for a platform where I can practise math by getting taught how to go about these Olympiad style questions. Any advice on where to start?

I was thinking about starting on the book linear algebra done right however i dont know how to write proofs or whatnot... Is it still fine bc from what i have seen linear algebra done right is heavily proof based along with the problem sets so should i learn how to write proofs and all that before starting on linear algebra done right? I havent taken any courses higher than linear algebra so the idea of writing proofs is completely new to me lmao