What’s the sum? 1 - 1/2 + 1/4 - 1/8 + 1/16...

One way to find the sum is by using the formula for a geometric series, but there’s an even more interesting visual approach!
In this video, we use the shaded area of a hexagon to reveal the answer. If you know of any other approach, let us know in the comments!
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[parthchatupale]

Though it's just an infinite GP its visual representation is awesome!!

[briogochill]

To calculate it you can write S this sum.
2*S=2-S
Therefore
3*S=2
S=2/3

[Couka]

S_n = S[for k = 0 to n] (-1/2)^k

-1 < -1/2 < 0 so, the sequence u_n = (-1/2)^n converges to 0

So you can write S_n = (1 - (-1/2)^n+1) / (1 - 1/2)
when n goes to infinite, lim S_n = 1 / (3/2) = 2/3

[l_I-I_l]

a+ar+ar²+...+arⁿ = a(1-rⁿ)/(1-r)
n -> inf => a+ar+ar²+... = a/(1-r) {-1<r<1}
let a=1, r=-1/2. then

[FIREFLY]

it's a basic G.P. question. Sum of infinite terms of a G.P.=first term/(1-Common Ratio). Here the common ratio is (-1/2). So sum is_ 1/(1-(-1/2))=2/3

[AnwarAdnan-ew]

Its geometric series with a=1 and r=-1/2
The infinit sum defined as
S=a/(1-r)
S=2/3

[ogbaconsfriend]

Geometry and physics are deeply intertwined, with numbers serving as the language that connects these two disciplines. Here's how numbers play a crucial role in understanding the relationship between geometry and physics:
Mathematical Descriptions of Physical Phenomena:
* Equations of Motion: In classical mechanics, equations of motion like Newton's second law (F = ma) describe the relationship between force (F), mass (m), and acceleration (a). These equations involve numerical values for force, mass, and acceleration, allowing us to calculate the motion of objects.

* Electromagnetism: Maxwell's equations unify electricity and magnetism, describing the behavior of electric and magnetic fields. These equations involve numerical values for electric and magnetic field strengths, charges, and currents.

* Quantum Mechanics: In the quantum world, equations like the Schrödinger equation describe the behavior of particles at the atomic and subatomic level. These equations involve complex numbers and probabilities, allowing us to calculate the likelihood of different outcomes in quantum experiments.

* General Relativity: Einstein's theory of general relativity describes gravity as a curvature of spacetime. This theory involves complex mathematical equations that relate the curvature of spacetime to the distribution of mass and energy. These equations involve numerical values for distances, masses, and time intervals.

Geometric Representations of Physical Concepts:
* Vectors: Vectors are mathematical objects that have both magnitude (size) and direction. They are used to represent physical quantities like force, velocity, and acceleration. The components of a vector are numbers that represent its magnitude and direction along different axes.

* Coordinates: Coordinate systems like Cartesian coordinates and polar coordinates are used to describe the position of points in space. These coordinates involve numerical values that represent the distances of points from the origin along different axes.

* Graphs: Graphs are visual representations of mathematical functions and data. They are used to plot physical quantities like position, velocity, and acceleration as a function of time. The axes of a graph are labeled with numerical values, and the points on the graph represent specific values of the plotted quantities.

Numerical Simulations:
* Computer Simulations: Computers can be used to solve complex mathematical equations and simulate physical phenomena. These simulations often involve numerical methods like finite element analysis and finite difference methods, which break down continuous problems into discrete steps involving numerical calculations.
* Data Analysis: Numerical data collected from experiments is analyzed using statistical techniques to extract meaningful information. These techniques involve calculations of averages, standard deviations, and correlations, which are all based on numerical values.
In Conclusion:
Numbers are the foundation of geometry and physics, allowing us to describe, quantify, and analyze the physical world. By using numbers to represent physical quantities, we can develop mathematical models that accurately describe the behavior of objects and systems. These models, in turn, allow us to make predictions, design experiments, and solve real-world problems.

[Cheeztoe]

Fun fact if you do the same thing but the denominators are odd numbers you will get one foruth of pi

[NyanCat-dnjb]

what i don't like about this animation is that he doesn't show proof that he is dividing the hexagon into half over and over again beyond 1/4. the equivalence of saying "Dude, trust me."

[Ricardo_S]

S=1-1/2+1/4-1/8+1/16-...=


1+(-1/2)S

S=1+(-1/2)S
S(1+1/2)=1
S(3/2)=1
S=2/3

[mohkh]

we can separate this out into two formulas

S1 = 1, 1/4, 1/16, 1/64...
S2 = -1/2, -1/8, -1/32...

in this way we can use this formula S = a/(1-r)
where S is the sum to infinity of a converging series a is the first term and r is the ratio of the 2nd term divided by the first term. Then when we find S1 and S2 we just add them together to get the final result

S1 = 1/(1-1/4) =4/3
S2 = (-1/2)/(1-1/4) = -2/3

S1 + S2 = 4/3 -2/3 = 2/3

[dvyanxu]

Cool thing

You can also try it with other shapes yourself and tey getting the same answer.

[Golden_Tortoise]

why is it a hexagon in the first place?

[HarryLarsson-bn]

why can't you just do this with a square

[reizinhodojogo]

i din't see that it was - then + then - then + and through it was 1 lol

[stephenliao]

It’s not helpful since if not having the result first only smart math guy can know where it stops without it animated.

[Geryboy]

that's a pseudo explanation, this is algebra, not geometry. you can't solve this with geometry.

[milesfreilich]

It can't be exactly ⅔. I bet there's a in there, somewhere.