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The coefficient of x^r (0 1) in the expression : (x+2)^n-1 +(x+2)^n-2 (x+1)+........+(x+1)^n-1 is ??? Please give detailed procedure. Thank You!!!

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!!!

Grade:11

2 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

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).

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Chaitanya,

 

A small correction.

It is nCr(2n-r - 1)xr, when you take common terms out.

 

All the best.

 

Regards,

Ashwin (IIT Madras).

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