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Haha, I guess the author was just lazy. I feel you though, there's been multiple times where I want to have a word with the crazy people who wrote the calculus book I have to use!

Btw, for the next step, looks like he just distributed the 27/n³ to each term in the parenthesees. I wasn't sure if you were wondering about that too.

Thanks so much hectictar i appreciate it. That's the exact bit i'm talking about. I looked at your post and saw it and said "That's it!"  Your post helps me out a lot.  I wonder why he didn't explain it all out in the book?  I thought that was the idea for the dummies books to completely break things down in easily understood language as Calculus can get complicated. Thanks again.

  Gary

I managed to get a screen shot of the page you're talking about:

I'm not claiming to understand what all this is... but I think I can show how he got from step 1 to step 2.

\(\frac{27}{n^3}\cdot\frac{n(n+1)(2n+1)}{6}+\frac{3}{n}\cdot n\)                        Multiply out the numerator.

\( \frac{27}{n^3}\cdot\frac{n(2n^2+3n+1)}{6}+\frac{3}{n}\cdot n \\~\\ \frac{27}{n^3}\cdot\frac{2n^3 + 3n^2 +n}{6}+\frac{3}{n}\cdot n\)

\(\frac{27}{n^3}\cdot(\frac{2n^3}{6} + \frac{3n^2}{6} +\frac{n}{6})+\frac{3}{n}\cdot n \\~\\ \frac{27}{n^3}\cdot(\frac{n^3}{3} + \frac{n^2}{2} +\frac{n}{6})+\frac{3}{n}\cdot n\)                Distribute the 1/6 and reduce.

\(\frac{27}{n^3}\cdot(\frac{n^3}{3} + \frac{n^2}{2} +\frac{n}{6})+3\)                          In the last term the n's cancel  :)

Hi mate, thanks. It's a Riemann sum or at least in that section of the book  and it's for calculating the area under a curve so it's integration. I'll try to write it out here, the first line of it anyway. He starts off with using the sigma summation sign and then on the next page he uses the integration sign and takes it to the limit of the function. I think basically he then takes the limit as infinity in order to get the exact area under the curve. 

27/n^{3 .}  n(n+1)(2n+1)/6 +  3.n/n   

That's the basic sum. The last bit is 3 times ndivided by n. The dot for multiplication is hard to see but i didn't want to use an x because it can cause confusion.  Now the second line in the book has a breakdown of the middle bit there, the n+1 bit, where he gets n3/3 + n²/2 + n/6 but that's the bit where he writes at the side that he didn't include all his work. So it's not broken down how he gets there and for a beginner it's just impossible trying to figure it out. I've looked on YT videos for the sum of the squares that the middle part of the sum concerns but most go into proving the theory or induction or something which is way above my head and too advanced for me. I'm just looking for a kind of simple breakdown if it's possible.  I think he omitted the explanation because he didn't want to have to write all the extra steps out!!

  Gary                

Lol! The hearts are kinda sweet though! 

Also on another note, I wonder whether it is good manners to up-thumb your own post? It seems a little weird at first.

But it is kind of like voting for yourself. All the candidates in an election should vote for themselves. That's what is expected and it makes sense.

Is it called the Simpson's Rule, Trapezoidal Rule, or Midpoint Rule? I'm just guessing...

A rule about intersecting chords says that:

XT * TY = AT * TB             Substitute in what each of these are.

4 * 6 = 2TB * TB               Simplify.

24 = 2(TB)²                       Divide both sides by 2.

12 = (TB)²                         Take the square root of both sides.

√(12) = TB                         Simplify.

2√3 = TB

AT = 2TB                           Substitute in 2√3 for TB.

AT = 2(2√3)                        Simplify.

AT = 4√3

AB = AT + TB                    Substitute in what we've found for AT and TB.

AB = 4√3 + 2√3                 Simplify.

AB = 6√3 ≈ 10.392

144 in base 62

Me too.

On another note.....I wonder if we can get rid of the cheesy heart symbol and replace it with something mathematical like

#   or  pi sign or ANYthing else.

1 degree a has 60 arc minutes    each minute has 60 arc seconds

360 x .39 x 60 x 60 = 505440 (arc)seconds

f(x) = -x³ + x² + x

To find f(-2), just replace every single " x " with " -2 ", then simplify.

f(-2) = -(-2)³ + (-2)² + (-2)

f(-2) = -(-8) + 4 + (-2)

f(-2) = 8 + 4 - 2

f(-2) = 10

Solve for x over the real numbers:
850 e^(0.055 x) = 195 e^(0.075 x)

850 e^(0.055 x) = 850 e^(11 x/200) and 195 e^(0.075 x) = 195 e^(3 x/40):
850 e^((11 x)/200) = 195 e^((3 x)/40)

Take the natural logarithm of both sides and use the identities log(a b) = log(a) + log(b) and log(a^b) = b log(a):
(11 x)/200 + log(850) = (3 x)/40 + log(195)

Subtract (3 x)/40 + log(850) from both sides:
-x/50 = log(195) - log(850)

Multiply both sides by -50:
Answer: | x = 50 log(850) - 50 log(195)= 73.61183954603077.......

How far someone can walk (or run) in one hour is a simple and common one.

x + y = 0.25                Subtract y from both sides.

x = 0.25 - y

x/y = 0.25                   Substitute (0.25 - y) in for x.

(0.25 - y)/y = 0.25       Distribute 1/y.

0.25/y - y/y = 0.25

0.25/y - 1 = 0.25         Add 1 to both sides.

0.25/y = 1.25              Multiply both sides by y.

0.25 = 1.25y               Divide both sides by 1.25.

0.2 = y

x = 0.25 - y                Substitute 0.2 in for y.

x = 0.25 - 0.2

x = 0.05

Just like Guest found. This is just an explanation.

1.496 x 10^11 m = 1 Astronomical Unit

1.496 x 1.7 x 10^11 / (3 x 10^8)  x  1 min/60 sec = 14.13  minutes

This is happening to me too.

I think it is just the administrator working on some things.

It should be up and running before too long :)

f^-1​(13 + 4) = 2f^-1​(13) + 1

f^-1​(17) = 2f^-1​(13) + 1

f^-1​(9 + 4) = 2f^-1​(9) + 1

f^-1​(13) = 2f^-1​(9) + 1

f^-1​(5 + 4) = 2f^-1​(5) + 1

f^-1​(9) = 2f^-1​(5) + 1

f(2) = 5, so f^-1(5) = 2

Substitute:

f^-1​(9) = 2(2) + 1 = 5

f^-1​(13) = 2(5) + 1 = 11

f^-1​(17) = 2(11) + 1 = 23

Seventeen.

arctan (1/4) =  14.04°

arccos (1/2)   = 60°

So

cos ( arctan (1/4)  + arccos (1/2) )   =

cos (14.04 + 60)  =

cos (74.04)   ≈ 0.275

x = 0.05

y = 0.20

Assuming it is 73 degrees.

\(\sin^{-1} (\frac{11}{12}*\sin(73^{\circ}))\)

Just type this whole thing into the calculator.

\(\sin^{-1} (\frac{11}{12}*\sin(73^{\circ})) \approx 61.236\)

Here is your solution bud!   

-a/(-3)+13 = 16

if  both b and c are in the denominator here is your answer

-a/-10 = 16

-a/-10 x 10 = 16 x 10

a = 160

now if b and c are seperate here is your solution

-a/(-3) + 13 = 16

-a/(-3) + 13 - 13 = 16 - 13

-a/(-3) = 3

negative numerator + denominator = positive

a/3 = 3

a/3 x 3 = 3 x 3

a=9

If I helped please leave a rating!

gcd(m,n) = 12     the two smallest numbers that are divisble by 12 that are different are 12 and 24

we will switch these areound for a smaller number m=24 n=12

10 times 24 = 240 and 15 times 12 is 180

240 pf; 2⁴ x 3 x 5     180 pf; 2² x 3² x 5    similar pf's = 2x2x3x5 which is equal to 60

60 is your answer!

if I helped please leave a rating

pf= prime factor

Unendlich (German for "never ending")