Question

Find the first three terms in the expansion , in ascending power of x , of (2+x)^6 and obtain the coefficient of x^2 in the expansion of (2+x-x^2)^6

please answer quickly​

Answer 1

The first 3 terms in the expansion of
`(2 + x)^(6)` , in ascending power of x are,

`64 , 192 * x^(1) {\textrm{  and  }}240 * x^(2)`

coefficient of
`x^(2)` in the expansion of
`(2+x - x^(2))^(6)` = (240 - 192) = 48

Explanation:

`(2+x)^(6)`

=
`\sum_(k=0)^(6)(6_{C_(k)} * x^(k) * 2^(6 - k))`

=
`6_{C_(0)} * x^(0) * 2^(6)  + 6_{C_(1)} * x^(1) * 2^(5) + 6_{C_(2)} * x^(2) * 2^(4)` + terms involving higher powers of x

=
`64 + 192 * x^(1) + 240 * x^(2)` + terms involving higher powers of x

so, the first 3 terms in the expansion of
`(2 + x)^(6)` , in ascending power of x are,

`64 , 192 * x^(1) {\textrm{  and  }}240 * x^(2)`

Again,

`(2+x - x^(2))^(6)`

=
`\sum_(k=0)^(6)(6_{C_(k)} * (2 + x)^(k) * (-x^(2))^(6 - k))`

Now, by inspection,

the term
`x^(2)` comes from k =5 and k = 6

for k = 5, the coefficient of
`x^(2)` is ,
`(-32) * 6` = -192

for k = 6 , the coefficient of
`x^(2)` is,
`6_{C_(2)} * 2^(4)` = 240

so, coefficient of
`x^(2)` in the final expression = (240 - 192) = 48