For natural numbers m , n, if [( 1 - y )^m( 1 + y )^n,, = 1 + a_1y + a_2y^2 + ..........] …

Question

For natural numbers $m$ , $n$,  if $${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........$$ and ${a_1} = {a_2} = 10,$ then $$\left( {m,\,n} \right)$$ is

Accepted Answer

$${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\,$$

= $$\left( {{}^m{C_0} - {}^m{C_1}y + {}^m{C_2}{y^2} + ....} \right)$$ -

$$\left( {{}^n{C_0} + {}^n{C_1}y + {}^n{C_2}{y^2} + ....} \right)$$

${a_1}$ = Coefficient of y = ${{}^n{C_1}}$ - ${{}^m{C_1}}$ = 10

$$ \Rightarrow $$ n - m = 10

${a_2}$ = Coefficient of y2

= $${}^n{C_2} + {}^n{C_1} \times {}^m{C_1} + {}^m{C_2} = 10$$

$$ \Rightarrow {{n\left( {n - 1} \right)} \over 2} - nm + {{m\left( {m - 1} \right)} \over 2} = 10$$

$$ \Rightarrow n\left( {n - 1} \right) - 2nm + m\left( {m - 1} \right) = 20$$

$$ \Rightarrow$$ (m + 10)(m + 9) - 2(m + 10)m + m(m - 1) = 20

$$ \Rightarrow$$ 90 + 19m + m2 - 2m2 - 20m + m2 - m - 20 = 0

$$ \Rightarrow$$ 70 - 2m = 0

$$ \Rightarrow$$ m = 35

$$\therefore$$ n = 10 + 35 = 45

Solutions · 1