Show that ratio of the coefficient of x10 in (1-x2)10 and the term independent of x in (x-2/x)10 is 1 - Maths - Binomial Theorem

Question

Show that ratio of the coefficient of x10 in
(1-x2)10 and the term independent of x in
(x-2/x)10 is 1 :32

Neha Sethi · Accepted Answer

Dear student

We know that general term in the expansion of a - bn isTr+1 = -1r 
  Crn   an-r 
 brSo, general term in the expansion of 1-x210 isTr+1 = -1r  10Cr 110-r 
 x2rTr+1=10Cr-1rx2rPutting r=5, we getT6=-15 
 10C5 x25 = -10C5 
 x10So, Coefficient of x10=-10C5general term in the expansion of x - 2x10 isTr+1 = -1r 10Cr x10-r × 2xrTr+1 = -1r 10Cr 
 2r 
 x10-2rPutting 10-2r=0, we getr=5So, term independent of x :T6=-15 10C5 
25=-32 10C5Hence their ratio=-10C5:-32 10C5=1 : 32

Regards

Show that ratio of the coefficient of x10 in (1-x2)10 and the term independent of x in (x-2/x)10 is 1 :32 .

Show that ratio of the coefficient of x¹⁰ in (1-x²)¹⁰ and the term independent of x in (x-2/x)¹⁰ is 1 :32 .