Fitzpatrick textbook question please help (BINOMIALS) (1 Viewer)

Kingportable · Mar 9, 2012

Can anyone help me? I am working on a question from John Fitzpatrick 3unit Text book. It is in Chapter 29a) question 50. please help it's really confusing me, i don't even know how to write the question in the forum.

nightweaver066 · Mar 9, 2012

Kingportable · Mar 9, 2012

Wow mate how did you do that and thanks =)

Drongoski · Mar 10, 2012

I've figured it out - but being a very slow typist, the prospect of keying in the solution is intimidating!

Drongoski · Mar 10, 2012

Solution to 2nd part.

$x^n + x^{n-1}(1+x) + x^{n-2} (1+x)^2 + \cdots + (1+x)^n = \frac {x^n [(\frac {1+x}{x})^{n+1} - 1]}{(\frac {1+x}{x})-1}$

$= (1+x)^{n+1} - x^{n+1} = \sum _{r=0}^{n+1} \binom {n+1} r x^r - x^{n+1} = \sum _{r=0} ^n \binom {n+1} r x^r$

being a geometric sum, 1st term xⁿ, common ratio (1+x)/x with (n+1) terms.

Its coeff of x^r $= \binom {n+1} r$

Now the coeff of x^r in:

$x^n + x^{n-1} (1+x) + \cdots + (1+x)^n = \binom n r + \binom {n-1}{r-1} + \cdots + \binom {n-r+1} 1+\binom {n-r} 0$

$= \binom n r + \binom {n-1}{r-1} + \cdots + \binom {n-r+1} 1 + 1 = \binom {n+1} r$

Hence the required sum $= \binom {n+1} r - 1$

Solution to 1st part given in separate post below.

girlworld_club · Mar 10, 2012

wow, that's amazing. are you even allowed to go to knox grammer if you don't go there.

Drongoski · Mar 10, 2012

During the Book Sale - a fairly big affair - yes; open to the public. All sorts of books.

Drongoski · Mar 10, 2012

1st part:

We have a geometric sum: 1st term = 1, common ratio = (1+x) no of terms = n+1

$= \frac {1[(1+x)^{n+1}-1]}{(1+x)-1} = x^{-1} [(1+x)^{n+1} - 1] = x^{-1}[ \sum _{r=0} ^{n+1}\binom{n+1} r x^r - 1]$

$= \sum _{r=1} ^{n+1} \binom {n+1} r x^{r-1}$ whose term in x^r is $\binom {n+1}{r+1} x^{r+1-1}$

Assuming n > r, we get tems in x^r in the binomial expansion of terms:

$(1+x)^n, (1+x)^{n-1}, (1+x)^{n-2}, \cdots (1+x)^r$ the sum of whose coeffs

$= \binom n r + \binom {n-1} r + \binom {n-2} r + \cdots + \binom r r = \binom {n+1}{r+1}$

Fitzpatrick textbook question please help (BINOMIALS) (1 Viewer)

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