User johnperkins

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Full name: John Perkins
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About: BS in math, MS in education, 27 years as a math tutor, 9 years as a background investigator.

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Cos^(2)15-sin^(2)15
Feb 5, 2016 by McCquabena Bannor
This is one of those problems that would be much easier to explain by any way other than writing.  It's easier to follow if you can see the problem being written out and have things pointed to.

Warning:  I think I'm right, but I could be wrong.

Look up a thing called "pascal's triangle."  It will make this much easier.

Let's do the expansion a bit...  Don't worry too much about this part, it's just a chart to use further down.  I listed more than we need below, but it's there to show a pattern.

(1+3x)^1 = 1 + 3x

(1+3x)^2 = 1 + 6x + 9x^2

(1+3x)^3 = 1 + 9x + 27x^2 + 27x^3

(1+3x)^4 = 1 + 12x + 54x^2 + 108x^3 + 81x^4

(1+3x)^5 = 1 + 15x + 90x^2 + 270x^3 + 405x^4 + 324x^5

(1+3x)^6 = 1 + 18x + 135x^2 + 540x^3 + 1215x^4 + 1944x^5 + 972x^6

If you were expanding (1+y)^n you would get things like 1+y, 1+2y+y^2, 1+3y+3y^2+y^3 and so on.  That's what Pascal's triangle gives us.

Look at the triangle, row 12 (1, 12, 66...).  If you wanted to expand (1+y)^12 you would get 1+12y+66y^2+220y^3...

But we're not expanding (1+y).  We're expanding (1+3x).  So instead of y, we have 3x.

As we expand (1+3x) we get 1+3x, 1+2(3x)+(3x)^2, 1+3(3x)+3((3x)^2)+((3x)^3), and so on.

The question asks about the coefficient of x^2 being 324, so it's asking about 324x^2.  We want to know when that 324 happens.

Remember we're not doing (1+x)^n, but (1+3x)^n.  That means we get BLAH(3x)^2 = BLAH(9x^2) = 324x^2.  (we don't know what BLAH is yet)

BLAH(9x^2) = 324x^2

Divide both sides by 9x^2

BLAH = 36

Look at the triangle again.

See how the left side is 1, 1, 1, forever?  That's the stuff that makes the 1 (all by itself).

See how the next side in is 1, 2, 3, 4, and so on?  That's the stuff that makes the number sitting next to the x (not the x^2, x^3, etc.).

See how the next side in is 1, 3, 6, 10, 15, 21, 28, 36, and so on?  That's the stuff that makes the number sitting next to the x^2 (not the x, x^3, x^4, etc.).

We care about the x^2, so we're interested in the 3rd step in on the left side (1, 3, 6, 10, ...).

The top row of the triangle (the 1 all by itself) is row 0.  The 1, 1 is row 1.  The 1, 2, 1 is row 2, and so on.

Remember BLAH = 36?  On the triangle, left side, 3rd step in, 9th row is 36.  The row number is n.  That means n = 9.

Answer:  n = 9



Another problem: If the coefficient of x^3 in the expansion of (1+4x)^n is 640, find n.

It's not x^3, it's (4x)^3 = 64x^3

640(x^3) = BLAH(64x^3)

Divide both sides by 64x^3

10 = BLAH

We're doing x^3, so it's the 4th step in from the left side.

Look for the 10.  It's in the 5th row. (remember the very top is row 0)

Answer:  n = 5.



Another problem:  If the coefficient of x^2 in the expansion of (1+2x)^n is 264, find n.

It's not x^2, it's (2x)^2 = 4x^2

264(x^2) = BLAH(4x^2)

Divide both sides by 4x^2

66 = BLAH

We're doing x^2, so it's the 3rd step in from the left side.

Look for the 66.  It's in the 12th row.

Answer:  n = 12.
Jan 4, 2016 by johnperkins
If the coefficient of x^2 in the expansion of (1+3x)^n is 324.find the value of n.Thanks
Jan 4, 2016 by McCquabena Bannor
Disclaimer:  I don't know how to do this problem.  I had to look it up.  Answer might be correct, might not.

Are you sure you meant Y^2=X^3+(A^n - B^n)X^2 - ( AB)^n and not Y^2=X^3+(A^n - B^n)X^2 - (( AB)^n)X ?

According to this page ( https://en.wikipedia.org/wiki/Modular_elliptic_curve ) that appears to be Y^2 = X(X+A^n)(X-B^n)

Note- your A and B are switched from the way it's described on that page.  It doesn't make any difference since we don't know what values A and B are.
Nov 22, 2015 by johnperkins
Please help me to solve this:
Let A, B, n to be whole numbers. What is the modular form of the elliptic equation:  Y^2=X^3+(A^n - B^n)X^2 - ( AB)^n
Nov 22, 2015 by lovelymath
Please help I do not understand.
Mar 2, 2015 by hiswife
A blue whale is swimming at an average rate of 3 miles per hour.Write a linear equation that gives the distance y in miles that the whale swims in x hour's . then graph the relationship
Mar 2, 2015 by hiswife
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