Surprising Patterns in the Square Numbers (1, 4, 9, 16…)


#81

for some thing before i haven’t seen this site, i am expecting [(n-1)2]+1 to be seen

anyway, i used that formula (that i invented or discovered by myself if someone had it) to get the difference before


#82

X


#83


#84

(-5X^3+8X^2+X+2)^2=???


#85

Triangular numbers anyone?

It wasn’t 'till much later that I learned about triangular numbers and the ‘Handshake Problem’ but I most certainly remember experiencing my first triangular number. This gave me a great experience for the way math can teach us about growth and how deceptive our intuition can be without reasoned calculation.

One particular kenpo class our instructor had a great idea for sparring. Everyone lineup. The first student takes his turn:
-first student spars 2nd until 3 points are scored (everyone else do 10 pushups)
-regardless of winner 1st student spars 3rd student (drop & give me 10 everyone else)
-continue until the 1st student spars last student
Now the second student gets his turn.
Everyone gets a turn!

If memory serves there were 12 of us that day. Bonus question: How many sparring matches? How many push ups? At an average 3 minutes per sparring match (with 1 round of 10 pushups being completed sooner) how long were we there?


#86

Thank you ,you help me a lots.I have stear on the web for thirty hours and still cannot think it instuitional.Could you tell me why the length of the arc equal to the"x" of e^xi ?forgive my Chinglish


#87

HI,the words:
’‘We apply i units of growth in infinitely small increments, each pushing us at a 90-degree angle. There is no “faster and faster” rotation - instead, we crawl along the perimeter a distance of |i| = 1 (magnitude of i).’’

I still cannot understand.If I think e^1 on the same way,then I get e^1=e(intuitively the length is e),but we get 1 radan(the length of arc is 1)from 'e^i '.
best regard


#88

very good


#89

Hi Dyron, great question (this is probably the wrong post though!).

When we have e^i, the base e is not the size or length: it’s the type of growth we have (growing continuously).

The exponent, i, has a size of 1, and modifies our growth to be a rotation. e^i is one radian around the circle, e^(2i) is 2 radians, and so on.


#90

I can waiting for to the school


#91

Did anybody else notice that the sum of squares is always 0,1,4,7,9?


#92

yes COLIN…i have noticed that…

moreover the sum digits in the difference between any two successive cube will
EITHER ‘1’ OR ‘7’…


#93

You might add something on percents, fractions and ratios. My daughter always had trouble. What was obvious to me was not to her. She needed a more visual approach. Such things as (but not limited to):
percents to/from fractions and decimal numbers
30% discount followed by a 20% discount on $100 is $56 not $50.
Doing the above problem as $100 * .7 * .8 = $56
A 50% decrease followed by a 50% increase yields 75% of the original value.
Adding and multiplying fractions


#94

What are the connections between
1, 1
2, 2
3, 4
4, 8
5, 16
6, 32
7, 64
8, 128
9, 256
10, 512


#95

if x,y,z are three successive numbers then,the cube of the middle number (y) is related to its neighbours as,

       y^3=[(x^2)*z)]+y^2+x.......


                              [THIS HOLDS GOOD FOR ANY NUMBER]

#96

6(x-1)+12 I figured this out on my free time without seeing this page. Simple.


#97

Delighted to have come across this article and interesting comments.
My 3rd grader just recently figured out the difference to the next square, which he thought of as n + (n +1), and I worked it out algebraically with him to show that it works for all cases. He will love the related observations above (though the calculus section will wait for another day :slight_smile: ).


#98

@seanoduill: Awesome, glad to hear you and your child enjoyed it!


#99

Ghee:
1=2^(1-1)
2=2^(2-1)
4=2^(3-1)
8=2^(4-1)
etc…


#100

Dear Kalid,
Great job Sir!
Your posts have had a great impact on the way I looked at Mathematics.
I thought math was just about remembering theorems and applying them in Problems and frankly speaking, the education system in India forced me to believe the same.
A genuine thanks!
*HUGS
Cheers!!!
-Paul