Trend in Square roots question [Curious Math Newb question]

In summary, the trend shows that the difference between the increases when a number is squared is linear, with each difference being 2 more than the previous one. This trend can be visualized using squares and can be represented mathematically as D_n = 2(n-1). This trend is also seen in a simpler progression, with a difference of 2 between each increase. The visual representation shows that the difference between successive increases is equal in size.
  • #1
nickadams
182
0
I noticed that 2 grows by 2 when it is squared, and 3 grows by 6 when it is squared, and 4 grows by 12, and 5 grows by 20... etc. etc.

So 3's increase when squared is 4 more than 2's increase when squared, and 4's increase when squared is 6 greater than 3's increase when squared, and 5's increase when squared is 8 bigger than 4's... etc. etc... <---The change from 4-5 is 2 bigger than the change from 3-4, and so on..



Why does this trend exist? Does it mean anything?


Thanks guys
 
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  • #2
Well let's see...
It helps to be able to write number relations as math vis:
the increase in a number n when you square it is given by - [itex]I_n=n^2-n[/itex]

the difference in the adjacent increases would be: [itex]D_n=I_n-I_{n-1}[/itex] which would be:[tex]D_n=\big [ n^2-n \big ] - \big [ (n-1)^2 - (n-1) \big ] = 2(n-1)[/tex]... which is linear.

That what you mean?
 
  • #3
Simon Bridge said:
Well let's see...
It helps to be able to write number relations as math vis:
the increase in a number n when you square it is given by - [itex]I_n=n^2-n[/itex]

the difference in the adjacent increases would be: [itex]D_n=I_n-I_{n-1}[/itex] which would be:[tex]D_n=\big [ n^2-n \big ] - \big [ (n-1)^2 - (n-1) \big ] = 2(n-1)[/tex]... which is linear.

That what you mean?

Oh thanks that was super helpful

The second part of my post was trying to say Dn - Dn-1 = 2 and that can be found by 2(n-1) - 2(n-1-1) = 2!
 
  • #4
Notice also the same linearity in the simpler progression 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9 ...

If you grasp visual concepts more easily, look at this:

squares.png


The multicoloured square on the left represents your original series, that on the right represents the simpler one mentioned above. In each case the two red squares are the difference between successive terms (in this case 62 and 72) - notice that in each large square the green and blue L-shapes are congruent.
 
  • #5
!

I find this observation interesting. It appears that as the number being squared increases, the increase in its square root also increases. This is due to the fact that when a number is squared, it is multiplied by itself, resulting in a larger number. This larger number then has a larger square root.

This trend is known as a quadratic relationship and is commonly seen in mathematics and science. It is also related to the concept of exponential growth, where the rate of increase becomes larger as the quantity being measured increases.

In terms of its significance, this trend can be useful in understanding and predicting various phenomena in the natural world. For example, in physics, the relationship between force and acceleration follows a quadratic trend. In economics, the relationship between production and cost can also exhibit a quadratic trend. By recognizing and studying this trend, we can gain a deeper understanding of the underlying principles and make more accurate predictions.

In conclusion, the trend in square roots that you have observed is a common and important mathematical relationship. It demonstrates the power of exponential growth and can be applied to various fields of study. I encourage you to continue exploring this concept and its applications.
 

What is a trend in square roots?

A trend in square roots refers to the pattern or direction that the square roots of a given set of numbers follow. This can be observed by plotting the square roots on a graph and looking for any consistent increase or decrease.

Why is it important to study trends in square roots?

Studying trends in square roots is important because it helps us understand the relationships between numbers and how they change. It also allows us to make predictions and solve problems that involve square roots.

What are some common trends in square roots?

Some common trends in square roots include linear, exponential, and periodic trends. Linear trends show a constant increase or decrease in square roots, exponential trends show a rapid increase or decrease, and periodic trends show a repeating pattern.

How do you identify a trend in square roots?

To identify a trend in square roots, you can plot the numbers on a graph and look for any consistent pattern or direction. You can also find the slope of the line connecting the points to determine if it is increasing or decreasing.

Are there any real-world applications of studying trends in square roots?

Yes, there are many real-world applications of studying trends in square roots. For example, in finance, understanding trends in square roots can help with calculating compound interest. In science, it can help with analyzing data and making predictions. In engineering, it can help with designing and building structures with specific dimensions.

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