Melody

Yes well, I hope you learn from it!

I want to help you learn, it was not my intention just to do your homework for you!!

the width of a square whose side is x cm, is increased by 5 cm and its length is decreased by 4 cm. How varies the area of the rectangle formed regarding side x of the original square?

Hello to you both :)

Omi, Where did the 400 come from?

Old area = x^2

New area = \((x+5)(x-4) = x^2+x-20\)

So the area has increased by x-20 units squared.

Of course, if x-20 is negative then the area will be smaller.  :)

-----------------------------

Hola a los dos :)

Omi, de dónde vino el 400 viene?

Área de Old = x ^ 2

Nueva área =\((x+5)(x-4) = x^2+x-20\)

Así que la zona se ha incrementado en x-20 unidades al cuadrado.

Por supuesto, si x-20 es negativa entonces el área será más pequeña. :)

I am sorry Omi I misread your question on my little screen :(

Is it a question?

9:9

=1:1

Multiply by 

(3+3i)/(3+3i)

Hello once again Jake   

Prove by induction     (n is  a nattural number)

\(1+\frac{1}{4}+\frac{1}{9}..........+\frac{1}{n^2}<2-\frac{1}{n}\\\)

Step 1

Prove true for n=1

LHS = 1/1^2 = 1

RHS = 2-1/1 = 1 LHS

Therefore true for n=1

Step 2

Assume true for n=k

\(1+\frac{1}{4}+\frac{1}{9}..........+\frac{1}{k^2}<2-\frac{1}{k}\\\)

Now prove true for n=k+1

that is, prove

\(1+\frac{1}{4}+\frac{1}{9}..........+\frac{1}{k^2}+\frac{1}{(k+1)^2}<2-\frac{1}{k+1}\\ 1+\frac{1}{4}+\frac{1}{9}..........+\frac{1}{k^2}+\frac{1}{(k+1)^2}<2-\frac{k(k+1)}{k(k+1)^2}\\ 1+\frac{1}{4}+\frac{1}{9}..........+\frac{1}{k^2}+\frac{1}{(k+1)^2}<2-\frac{k^2+1}{k(k+1)^2}\\ \\ LHS<(2-\frac{1}{k})+\frac{1}{(k+1)^2}\\ LHS<2-\left(\frac{1}{k}-\frac{1}{(k+1)^2}\right)\\ LHS<2-\left(\frac{(k+1)^2}{k(k+1)^2}-\frac{k}{k(k+1)^2}\right)\\ LHS<2-\left(\frac{k^2+k+1}{k(k+1)^2}\right)\\ LHS<2-\frac{k^2+1}{k(k+1)^2}-\frac{k}{k(k+1)^2}\\ therefore\\ LHS<2-\frac{k^2+1}{k(k+1)^2}\\ LHS<2-\frac{1}{k+1}\\ \)

Therefore, if true for n=k then it must be true for n=k+1

Step 3

Since true for n=1, it must be true for n=2

Since true for n=2 it must be true for n=3, n=4,n=5, etc.  

Therefore true for al n   where   \(n\in N\)

Fri 22/10/15

1)  This logic question has sparked a lot of interest.  Thanks CPhill, Guest and Heureka.

       http://web2.0calc.com/questions/p-or-p-and-q

2)  Proof by Induction

     http://web2.0calc.com/questions/how-do-you-proove-this-inductuction

3) Find all zeros of a polynomial    Thanks Heureka

    http://web2.0calc.com/questions/math_9780

4)  Longitude and latitude.     Thanks Heureka and guest

    http://web2.0calc.com/questions/halp-me-plz-bb

5) Find potential energy.      Thanks Rom

    http://web2.0calc.com/questions/a-roller-coaster-has-4-cars-with-a-total-mass-of-1-000-each-car-holds-6-people-with-an-average-weight-of-150-n-how-much-potential-energy-do-the-cars-have

6) Interesting trig manipulation.  Thanks Heureka and guest.

  http://web2.0calc.com/questions/z-sin-3-x-cos-5-x-dx

Thanks Jake,

Welcome to the forum.    

Your English is very good :)  

@@ What is Happening?  [Wrap4]    TRhurs 22/10/15     Port Macquarie, Australia Time 10:48pm   ♪ ♫

Hi,

Again, Iam not very aware of what was posted today, I was not on much.   I give a hearty thankyou to all our answerers and I am sorry if your question was one that did not get answered.  It is impossible to answer all question athough I think most got at least some kind of a response. :/

Here are a few interesting questions/answers.  

1)  Trigonometry limit.     Thanks Melody and Alan.  PLUS another extravaganza answer by Heureka. Thanks         http://web2.0calc.com/questions/aghaee/new#edit

2) Good puzzle question.  Thanks guest :)

    http://web2.0calc.com/questions/reversals

3)  Physics, Magnitude adn acceleration    Thanks Alan

    http://web2.0calc.com/questions/cylindrical-components

4) Tricky integration.    Not Answered        One of xvxvxv's

    http://web2.0calc.com/questions/integration_7025

5)  Finding the inverse of a matrix.    Thanks Heureka and Rom

   http://web2.0calc.com/questions/find-the-inverse-of-the-matrix_1

6) solving a system of equations using matrices.   Thanks Alan.

    http://web2.0calc.com/questions/solve-the-system-using-the-inverse-method

NOTE: There are more matrix questions on page 2267   (25 per page)

If you want to comment an any issues please do so.    

Enjoy the rest of your day :) 

Melody    ♪ ♫

----------------------------------------------------

Lantern thread:

http://web2.0calc.com/questions/wrap-lantern-thurs-22-10-15

Thurs 22/10/15   .....

1)  Trigonometry limit.     Thanks Melody and Alan.  PLUS another extravaganza answer by Heureka. Thanks         http://web2.0calc.com/questions/aghaee/new#edit

2) Good puzzle question.  Thanks guest :)

    http://web2.0calc.com/questions/reversals

3)  Physics, Magnitude adn acceleration    Thanks Alan

    http://web2.0calc.com/questions/cylindrical-components

4) Tricky integration.    Not Answered        One of xvxvxv's

    http://web2.0calc.com/questions/integration_7025

5)  Finding the inverse of a matrix.    Thanks Heureka and Rom

   http://web2.0calc.com/questions/find-the-inverse-of-the-matrix_1

6) solving a system of equations using matrices.   Thanks Alan.

    http://web2.0calc.com/questions/solve-the-system-using-the-inverse-method

   NOTE: There are more matrix questions on page 2267   (25 per page)

2^2=2×2=4