Question

The first three terms in binomial expansion of (p-q)∧m, in ascending power of q, are denoted by a, b, and c.How to show that b∧2/ac = 2m/m-1?

Answer 1

See the proof below

Explanation:

The binomial expansion is

`(p-q)^m=p^m-((m),(1))p^(m-1)q+((m),(2))p^(m-2)q^2+..........`

`(p-q)^m=p^m-mp^(m-1)q+(m(m-1))/2p^(m-2)q^2+..........`

So, we have

`a=p^m`

`b=-mp^(m-1)q`

`c=(m(m-1))/2p^(m-2)q^2`

Therefore,

`b^2/(ac)=(-mp^(m-1)q)^2/((p^m)*((m(m-1))/2p^(m-2)q^2))`

`=(m^2p^(2(m-1))q^2)/((m(m-1))/2*p^(2m-2)q^2)`

`=(2m)/(m-1)`

`QED`