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)*[a^n-1 + a^n-2b +...... + b^n-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 ⁿC_rx^r2^n-r - ⁿC_rx^r = (ⁿC_r2^n-r - 1)x^r.

That is the answer.

Hope it helps.

Wish you all the best.

Regards,

Ashwin (IIT Madras).

Hi Chaitanya,

A small correction.

It is ⁿC_r(2^n-r - 1)x^r, when you take common terms out.

All the best.

Regards,

Ashwin (IIT Madras).