Find the coefficient of x^12 in (1-x^2)^-5 what can you set about the coefficient of x^17

Accepted Solution

Answer with explanation:The expansion  of   [tex](1+x)^n=1 + nx +\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+......[/tex]where,n is a positive or negative , rational number.Where, -1< x < 1Expansion of  [tex](1-x^2)^{-5}=1-5 x^2+\frac{(5)\times (6)}{2!}x^4-\frac{5\times 6\times 7}{3!}x^6+\frac{5\times 6\times 7\times 8}{4!}x^8-\frac{5\times 6\times 7\times 8\times 9}{5!}x^{10}+\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}x^{12}+....[/tex]Coefficient of [tex]x^{12}[/tex] in the expansion of [tex](1-x^2)^{-5}[/tex] is         [tex]=\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}\\\\=\frac{15120}{6\times 5 \times 4\times 3 \times 2 \times 1}\\\\=\frac{151200}{720}\\\\=210[/tex]⇒As the expansion [tex](1-x^2)^{-5}[/tex] contains even power of x , so there will be no term containing [tex]x^{17}[/tex].