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The coefficient of x^r (01) in the expression :
(x+2)^n-1 +(x+2)^n-2 (x+1)+........+(x+1)^n-1 is ???
Please give detailed procedure.
Thank You!!!
Hi Chaitanya,
Let x+2 = a, and x+1 = b.
We know (a-b)*[an-1 + an-2b +...... + bn-1] = an - bn
So the given expression is say S.
We have [(x+2) - (x+1)]*S = (x+2)n - (x+1)n.
Hence S = (x+2)n - (x+1)n which is the given expression.
The co-efficient of x^r in that expression would be nCrxr2n-r - nCrxr = (nCr2n-r - 1)xr.
That is the answer.
Hope it helps.
Wish you all the best.
Regards,
Ashwin (IIT Madras).
A small correction.
It is nCr(2n-r - 1)xr, when you take common terms out.
All the best.
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