[MOVED] Arithmetic in binomial coefficents

[Irfan]

Given that the coefficients of \(\displaystyle x^{r-1},\, x^r,\, x^{r+1}\) in the expansion of \(\displaystyle (1\, +\, x)^n\) are in arithmetic sequence, show that \(\displaystyle n^2\, +\, 4r^2\, -\, 2\, -\, n(4r\, +\, 1)\, =\, 0\)
Hence find three consecutive coefficients of the expansion of \(\displaystyle (1\, +\, x)^{14}\) which form an arithmetic sequence.

I don't even know how to start calculate this. I can only think of searching manually in the pascal triangle for an arithmetic sequence.
Does anyone have a clue to calculate this other than my way?

 

[Ishuda]

Given that the coefficients of \(\displaystyle x^{r-1},\, x^r,\, x^{r+1}\) in the expansion of \(\displaystyle (1\, +\, x)^n\) are in arithmetic sequence, show that \(\displaystyle n^2\, +\, 4r^2\, -\, 2\, -\, n(4r\, +\, 1)\, =\, 0\)
Hence find three consecutive coefficients of the expansion of \(\displaystyle (1\, +\, x)^{14}\) which form an arithmetic sequence.

I don't even know how to start calculate this. I can only think of searching manually in the pascal triangle for an arithmetic sequence.
Does anyone have a clue to calculate this other than my way?

The r^th term in the expansion of (1+x)ⁿ is the binomial coefficient _nC_r, see
http://en.wikipedia.org/wiki/Binomial_coefficient
for example.

Also, if A*a, A*b, and A*c are in arithmetic sequence, so are a, b, and c. So find three successive terms in the expansion of (1+x)ⁿ which are in arithmetic sequence, i.e. from the post by Denis, find an r so that

_nC_r+1 = _nC_r + d
and
_nC_r+2 = _nC_r + 2 d

and see where that leads you. If you get stuck and want more help, post again and show what you have done.

 

[Irfan]

Ishuda do you mean i have to expand it until I find an arithmetic sequence in its coefficients?

 

[Ishuda]

Ishuda do you mean i have to expand it until I find an arithmetic sequence in its coefficients?

Not the complete expansion. Look at the general coefficients _nC_r-1, _nC_r, and _nC_r+1. The common factor A which I mentioned in the earlier post would be something like a factor of \(\displaystyle \frac{n!}{(r+1)! (n-r+1)!}\) although that may not be quite it. The remaining terms (the a, b, and c) would be something like mixed quadratic/linear expressions in n and r which would be much easier to deal with. Maybe those expressions would give you a clue and you can get an r value in terms of n. That is
c = b + d = b + (b-a) = 2b - a
so what is
c - 2b + a

BTW: I only played around with this a little and I wasn't convinced that the arithmetic progression existed. I may have done something wrong though but then we weren't asked to prove they existed, just that if they did then the given equation followed.

Oh, and another way to approach this is to find solutions to the quadratic. Either r in terms of n, i.e treat the function as a quadratic in r and solve for the r values in terms of n. The discriminate [(n+2)?] must be a perfect square for one of the the solutions to be an integer and one of the solutions must be an integer. Or maybe the other way around (solve for n in terms of r). If the solution is 'moderately easy', check to see if _nC_r-1, _nC_r, and _nC_r+1 are in arithmetic progression.

One last comment; I have been known to go the (very) long way around in trying to solve a problem and not have success.

Edit:After playing around a bit more, I would definitely go the first method.

 

[Deleted member 4993]

Given that the coefficients of \(\displaystyle x^{r-1},\, x^r,\, x^{r+1}\) in the expansion of \(\displaystyle (1\, +\, x)^n\) are in arithmetic sequence, show that \(\displaystyle n^2\, +\, 4r^2\, -\, 2\, -\, n(4r\, +\, 1)\, =\, 0\)
Hence find three consecutive coefficients of the expansion of \(\displaystyle (1\, +\, x)^{14}\) which form an arithmetic sequence.

I don't even know how to start calculate this. I can only think of searching manually in the pascal triangle for an arithmetic sequence.
Does anyone have a clue to calculate this other than my way?

Since _nC_r-1, _nC_r and _nC_r+1 are consecutive terms of an Arithmetic Progression - we have: _nC_{r-1 + n}C_r+1 = _{2 * n}C_{r .... Now slog through the algebra and you'll get there (I did!!)}