Show that the coefficient of xn in the expansion of (1−x)2−cxx is n{1+⌊3n2−1c+⌊5(n2−1)(n2−4)c2+⌊7(n2−1)(n2−4)(n2−9)c3+.....}.

Question

Show that the coefficient of xn in the expansion of (1−x)2−cxx​ is n{1+⌊3n2−1​c+⌊5(n2−1)(n2−4)​c2+⌊7(n2−1)(n2−4)(n2−9)​c3+.....}.

Filo · Accepted Answer

The co efficient can be expressed as co efficient of

x^{n - 1}

in the expansion of

\frac{1}{( 1 - x ) ^{2} - c x}

or

\frac{1}{( 1 - x ) ^{2}} {1 - \frac{c x}{( 1 - x )}}^{- 1}

Expanding by Binomial Theorem, the last expression becomes,

\frac{1}{( 1 - x ) ^{2}} + \frac{c x}{( 1 - x ) ^{4}} + \frac{c ^{2} x ^{2}}{( 1 - x ) ^{6}} + \frac{c ^{2} x ^{3}}{( 1 - x ) ^{8}} + ....

and we have to pick out from the expansions of

(1 - x)^{- 2}, (1 - x)^{- 4}, (1 - x)^{- 6}, ....

co efficient of;-

= n + \frac{c ( n - 1 ) n ( n + 1 )}{3 !} + \frac{c ^{2} ( n - 2 ) ( n - 1 ) n ( n + 1 ) ( n + 2 )}{5 !} + ....

= n {1 + \frac{( n ^{2} - 1 )}{3 !} c + \frac{( n ^{2} - 1 ) ( n ^{2} - 4 )}{5 !} c^{2} + ....}

Question Text	Show that the coefficient of $x^{n}$ in the expansion of $\frac{x}{( 1 - x ) ^{2} - c x}$ is $n {1 + \frac{n ^{2} - 1}{⌊ 3} c + \frac{( n ^{2} - 1 ) ( n ^{2} - 4 )}{⌊ 5} c^{2} + \frac{( n ^{2} - 1 ) ( n ^{2} - 4 ) ( n ^{2} - 9 )}{⌊ 7} c^{3} + .....}$ .
Answer Type	Text solution:1
Upvotes	150

Show that the coefficient of $x^{n}$ in the expansion of $\frac{x}{( 1 - x ) ^{2} - c x}$ is $n {1 + \frac{n ^{2} - 1}{⌊ 3} c + \frac{( n ^{2} - 1 ) ( n ^{2} - 4 )}{⌊ 5} c^{2} + \frac{( n ^{2} - 1 ) ( n ^{2} - 4 ) ( n ^{2} - 9 )}{⌊ 7} c^{3} + .....}$ .