Respuesta :
Answer:
The first 3 terms in the expansion of [tex](2 + x)^{6}[/tex] , in ascending power of x are,
[tex]64 , 192 \times x^{1} {\textrm{ and }}240 \times x^{2} [/tex]
coefficient of [tex]x^{2}[/tex] in the expansion of [tex](2+x - x^{2})^{6}[/tex] = (240 - 192) = 48
Step-by-step explanation:
[tex](2+x)^{6}[/tex]
= [tex]\sum_{k=0}^{6}(6_{C_{k}} \times x^{k} \times 2^{6 - k})[/tex]
= [tex]6_{C_{0}} \times x^{0} \times 2^{6} + 6_{C_{1}} \times x^{1} \times 2^{5} + 6_{C_{2}} \times x^{2} \times 2^{4}[/tex] + terms involving higher powers of x
= [tex]64 + 192 \times x^{1} + 240 \times x^{2} [/tex] + terms involving higher powers of x
so, the first 3 terms in the expansion of [tex](2 + x)^{6}[/tex] , in ascending power of x are,
[tex]64 , 192 \times x^{1} {\textrm{ and }}240 \times x^{2} [/tex]
Again,
[tex](2+x - x^{2})^{6}[/tex]
= [tex]\sum_{k=0}^{6}(6_{C_{k}} \times (2 + x)^{k} \times (-x^{2})^{6 - k})[/tex]
Now, by inspection,
the term [tex]x^{2}[/tex] comes from k =5 and k = 6
for k = 5, the coefficient of [tex]x^{2}[/tex] is , [tex](-32) \times 6[/tex] = -192
for k = 6 , the coefficient of [tex]x^{2}[/tex] is, [tex]6_{C_{2}} \times 2^{4}[/tex] = 240
so, coefficient of [tex]x^{2}[/tex] in the final expression = (240 - 192) = 48
